GRAMMAR 



INFINITE FORMS. 









EDINBURGH, 
PRINTED BY OLIVER & BO I'D. 
TWEEDDALE-COURT. 



GRAMMAR 



INFINITE FORMS; 



THE MATHEMATICAL ELEMENTS OF ANCIENT 
PHILOSOPHY AND MYTHOLOGY. 



By WILLIAM HOWISON. 



ramos annosaque brachia pandit 

Ulmus opaca, ingens : quam sedem somnia vulgo 
tenere ferunt, foliisque sub omnibus hserent. 



EDINBURGH : 

PUBLISHED BY 

OLIVER & BOYD, TWEEDDALE-COURT ; 

SOLD ALSO BY 
G. & W. B. WHITTAKER, LONDON. 




1823. 



e*-~ -? 



I 






I 



PREFACE. 



The purpose of this treatise is, in the first place, 
to recall from oblivion the knowledge of those 
elementary powers of mathematical form which 
were the basis of all the fables belonging to that 
ancient system of philosophy and poetry which is 
now called mythology. These are found as twelve 
successive steps in one continued deduction, which 
takes in all the simple or uncompounded powers 
of mathematical form ; and, although the know- 
ledge of their deductive order has been lost or for- 
gotten during the intervening ages, it is shewn, by 
many existing traces, to have been well understood 
at one time by the Greeks and other cotempo- 
rary nations. It was the chief source of the truth 
of their conceptions and the correctness of their 



VI 



taste, which always approached as near as possible 
to abstract rule. 

The present treatise is called a Grammar, be- 
cause it shews the order and syntax of these powers, 
and defines the qualities peculiar to each. But 
the second part relates to the modes of composition 
which are derived from the union of powers, and 
which modes of composition are also found as steps 
in a regular series. 

The author of the following pages, in pursuing 
these abstractions, may seem to be exerting an 
useless ingenuity, and spending labour in vain, for * 
the purpose of resuscitating antiquated and scarce- 
ly amusing fables. He is of opinion, however, 
that none of these were originally contrived at 
random, or for commemorating particular events, 
but were originally intended for a totally different 
purpose, and are valuable memorials of the study 
of abstract truths, which cannot even now be ex- 
pressed by a more convenient set of symbols. 
But, in this treatise, the compositions, and the 



Vll 



modes of being which result from them, are fre- 
quently illustrated by references to modern na- 
tions, characters, events, or works of genius, in 
which the natures belonging to them seem to have 
been exemplified. And, although the coincidences 
pointed out may perhaps sometimes awaken a 
sense of the ludicrous, the author of this treatise 
professes to have no intention of insulting any 
department by the illustrations. It cannot but 
serve a good end to exhibit the severe forms of 
abstract truth without any accommodation to the 
feelings of mankind ; who may be accused of what, 
in the present condition of society, must be con- 
sidered as an excessive respect for human nature. 



PREFACE. 



The purpose of this Treatise is to help to deter- 
mine an important question which occurs in phi- 
losophy, and which is, Whether the number of 
hinds or modes of being, exemplified in nature, be 
limited or not ? It is evident that each kind may 
have subordinate classes, but these cannot exist 
apart from it. This Treatise is intended to shew 
that every kind, which is really apart from others, 
must be founded on some simple power, existing 
in abstract idea, that is to say, having a mathema- 
tical existence. Now, the number of simple ma- 
thematical powers (which are capable of flux and 
progression) will be found to be limited ; and, 
therefore, the number of kinds, or modes or be- 
ing, which are founded on them, must also be li- 
mited. 



VI 



The author of the following pages has been led 
to suppose that the whole system of Grecian my- 
thology had a mathematical origin, and that the 
beautiful differences of character, in the chief di- 
vinities, were the same as those of the simple ma- 
thematical powers which are capable of flux and 
progression. Thus the attributes of each divinity 
became entirely distinct from those of the rest ; 
and each inhabitant of Olympus assumed a cha- 
racter so clearly and strongly discriminated as 
never afterwards to be forgotten, either among 
barbarians, or more civilized generations. 

Part First of the following Treatise relates to the 
simple mathematical powers which are capable of 
flux ; and these are found deductively as steps in 
a series, wherein each power presupposes that 
which is immediately anterior. In this mathe- 
matical theorem, the first steps are of the most 
obvious nature ; but the rest are such as not to 
occur so readily to the mind. The deductive 
order of these powers, however, must undoubtedly 



J 



Vll 

have been studied and perceived by the ancients, 
although it has been overlooked or allowed to fall 
into oblivion during the intervening ages. 

Part Second of the following Treatise is an in- 
quiry into the mathematical flow or progression of 
compositions which, in number and difference of 
kind, correspond with the simple powers. In the 
latter, there is merely the flux of quantity and 
proportion which takes place in the same form 
prolonged and continued. But composition pro- 
duces a new beauty, which is the flux of kind, 
or the progressive exemplification of the same ge- 
neric power as it changes through an endless series 
of particular instances or acts. This in fact is the 
same as the nature of animated beings, and corre- 
sponds with mental life or activity. In this de- 
partment, ancient fable and imagination chiefly 
delighted to luxuriate; and mythology, besides 
particularizing the attributes of the chief divinities, 
as elementary powers, farther abounded with the 
characters of heroes, demigods, and local powers, 



Vlll 

all intended to express the kinds of intellectual life 
or mental sensation. These were, at the same 
time, really exemplified in the nations and races 
of mankind, to which, as well as to works of 
genius, the author of the following Treatise has 
frequently referred for illustration. 



CONTENTS. 



PAUT I. 

ON THE MATHEMATICAL ELEMENTS. 



PAGE 

Chap. I. On Contingency or External Relation . , 1 
II. On Fixed Position, Form, and Measure- 
ment . . . . . 5 

III. On Continuity and Abstract Rule . . 9 

IV. On Contrast, or Difference of Rule . 13 
V. On Definition, or the Limitation of Ab- 
stract Rules . . . . 16 

VI. On Indefinite Volume or Diffusion . 20 
VII. On Terminated Volume . . 24 

VIII. On the Prospective Relation of Solid 

Figure to Motion ... 28 

IX. On acquired Motion ... 31 
X. On Growing, or the Assumption of Ex- 
traneous Forces ... 36 
XI. On Composition ... .40 
XII. On Comprehension and Imagination . 46 



PART II. 

ON THE MODES OP COMPOSITION. 



PAGE 

Chap. I. On the Powers of Gradation, or the Com- 
parison of different Finite Series. 
Orion or Bacchus ... 49 
II. On the Power of Searching out the differ- 
ent Forms of the Saturnian Hyperbola. 
Hercules and the Muses . . 60 

III. On the Application of Straight Lines to 
the Series of Hyperbolic Branches. Pan 
or Msculapius .... 67 
IV. On the Retrogressions of the Hyperbola. 
Section I. On the Infinitude of a single 

Curve. Pelops or Proserpina 72 
II. On a Retrogressive Series of 
Branches, or the Power of 
Search into Antiquities. 
Pluto or Adonis . 81 

III. On the Evolution of Diminish- 
ing Hyperbolic Branches. 
Perseus ... 88 

IV. On the Retrogression of the 
Hyperbola through a Series 
of Inconsecutive Forms. 
Vertumnus or the Fates 94 






XI 

PAGE 

Chap. V. On the Interminable Forms of Trans- 
verse Progression. 
Section I. On the Powers of Continu- 
ity and Syntax. Erich- 
thonius or Hyperion . 100 
II. On the Deduction of 
Composite Hyperbolic 
Branches. Theseus . 107 
Chap. VI. On the Progression of Double Series 
of Hyperbolic Branches. 
Section I. On the Powers of Fluctua- 
tion. Pollux or Eridanus 117 
II. On the continued Relation 
of two Simple Curves. 
Castor or Bellerophon 123 
Chap. VII. On the Powers of Collocation, or the 
Distribution of Finite Parts of the 
Hyperbola. Geryon or Silvanus . 131 



PART I. 

ON THE MATHEMATICAL ELEMENTS. 



CHAP. I. 

ON CONTINGENCY, OR EXTERNAL RELATION. 

From the entire and absolute sameness of a single 
position no other mathematical idea is generated ; 
and, therefore, the difference of positions must be 
ranked as the first creative power, since it gener- 
ates something else, namely, relation passing to and 
fro between the positions. This cannot be fixed 
or made perceptible to the senses ; but its exist- 
ence is recognised by intellect. The contrariety 
of points is the origin of adventitiousness ; because 
external relation is not implied in the nature of 



either point taken separately. To the god Uranus 
or Caslus, among the eldest deities, was attributed 
the power of connecting all things ; and his char- 
acter signified merely relation. To the same 
power must also be ascribed the simplest and most 
original idea of motion, as the transition between 
different points ; and a moving body partakes of 
the character of Caelus, by producing a connexion 
between those places which it leaves and those 
into which it comes. From motion comes fortune, 
which belongs to existences capable of changing 
their places, and creating novelty of circumstances. 
Among the twelve chief heathen divinities, this 
power belonged to Juno ; and, among the tribes 
of Israel, the same character seems to have belong- 
ed to the tribe descended from Reuben, the eldest 
son of Jacob. The power of Juno, originating 
from the difference of units, becomes also the same 
as accession, and may be extended to the increase 
of numbers by continued accession from without ; 
because her power, being that of adventitiousness, 
can never be supposed to stand still, or be limited 
to any particular number. The arithmetical series 
follows out the differences of units or equal num- 



bers, and assumes a new proportion at each step. 
To Juno must also be ascribed the power of fluc- 
tuation, as arising from the alteration of distances ; 
because the relation of points or units continues 
to subsist, however much their distances may be 
changed, or into whatever order they be thrown. 
But any form, having a certain order and arrange- 
ment of parts, cannot be thus shaken or dis- 
composed without losing its identity. The kind of 
feeling, therefore, which must be ascribed to Juno 
is that of temporary states and fluctuations. Thus, 
the dancing of Salome procured from Herod a 
boon which occasioned the decapitation of St John 
the Baptist. To Juno, besides the fluctuations of 
the feelings, may also be ascribed pilgrimage in re- 
gard to the meeting of external novelty. Actuated 
by a strong feeling of the changeableness of the 
world, and its incessant agitation, the ancient 
astrologers betook themselves to watch the revolu- 
tions of the planets, and observe their conjunctions, 
of which they endeavoured to find the connexion 
with the fortunes of mankind in the world beneath. 
To the same class with Caelus and Juno may be 
referred the muse Urania. Juno was also my- 



thologically identified with the air, as the element 
most capable of mutability and agitation. From 
this power, however, the deduction of mathemati- 
cal elements must begin. 



CHAP. II. 

ON FIXED POSITION, FORM, AND MEASUREMENT. 

It is evident that more than two points, viewed 
together, may either be considered as capable of 
freely changing their order and distances, or they 
may be fixed to certain places, by having their 
distances compared, and a certain proportion esta- 
blished among these distances. From the com- 
parison of distances, therefore, comes fixed posi- 
tion. One point may revolve round another and 
preserve always the same distance; but a third 
point cannot revolve round one of the two first, 
without changing the proportion of the distances 
between it and the other two. The comparison 
and proportion of distances is also the origin of 
form ; although the different measurements which 
meet in one form may be innumerable. Among 
the twelve chief heathen divinities, form and fixed 
position was signified in the character of Jupiter ; 



6 

who, in reference to fixed position, was supposed 
to let down from his throne a great golden chain, 
on which all things depended. In another point 
of view, Jupiter, being only the offspring of Saturn 
and Rhea, was of more recent birth than many 
other divinities known in mythology; but, from 
his character relating to a centre, he was hypo- 
thetically called the father of gods and men. 
Among the twelve tribes of Judea, this power 
must have belonged to Simeon. In this power 
also the number of points employed must be sup- 
posed to be limited; because the characteristic 
of Jupiter is stability, which rejects the change 
produced by incessant external accession when 
continued, as in the power of Juno. And the 
only kind of flow or progression, which is appro- 
priate to a system of fixed proportions, is that 
which takes place in time ; as when a certain form 
of rhythmus or musical measure, having been gone 
through, is again begun, and repeated for ever in 
a circle. Thus, in poetry, each stanza, or some- 
times each verse, begins the same measure anew ; 
and, in this kind of progression, no new form is 
discovered, except in the increase of the whole 



series. The simple inhabitants of a pastoral coun- 
try are averse to think that there can be any ad- 
vantage not implied in the mode of existence which 
they already enjoy; and thus they are induced 
to harden themselves in a sort of stupidity, and to 
reject all new affections. The power of Jupiter 
must ever be opposed to accident and fluctuation, 
and to every sort of feeling not resulting from 
what is already possessed ; and, on the contrary, 
it rejoices in seeing the same cycle of events re- 
peated. To the same class with Jupiter must also 
be referred the three original muses, who were the 
daughters of Cselus and Terra, and, according to 
ancient authors, were known before the nine 
daughters of Jupiter and Mnemosyne. The three 
earlier muses may have signified the three generic 
powers either of musical rhythmus or verse ; name- 
ly, first, the proportion of quantity, such as that 
of a long to a short ; secondly, their order or dis- 
tribution ; and, thirdly, their number, from whence 
the extent of the whole form or measure to which 
they belong. To Jupiter must also be ascribed 
the power of acceleration and of producing the 
same form on a different scale ; for it is evident 



8 

that a system of points, having established dis- 
tances, is capable of revolution round any one of 
the points assumed as a centre. Those points, 
which have unequal distances from it, will revolve 
in concentric circles, and so trace the same form 
on different scales ; and the power of acceleration 
results from the more distant points being forced 
to revolve with the greater speed, to preserve their 
position or distance in regard to those nearer the 
centre. To the power of Jupiter, therefore, may 
be referred the low T er kinds of mental vehemence 
and impetuosity. Among tribes of barbarians, 
the feeling of rotatory motion is used for every 
purpose, whether of religion or of war ; and, being 
the lowest sort of mental enjoyment, it is given as 
a free inheritance to all created beings. To the 
same class with Jupiter may also be referred the 
goddess Themis, that is justice, which is 'equality, 
the simplest kind of proportion found by the com- 
parison of quantities. Revenge is counted the 
chief virtue among natures which are fixed, and 
incapable of progressive change or expansion. 



CHAP. III. 

ON CONTINUITY AND ABSTRACT RULE. 

The third mathematical element is continuity, by 
which is meant not relation passing to and fro be- 
tween different points, but a line, whose parts 
are fixed and remain in their places, and whose 
continuity may be prolonged to any extent. A 
line, however, can never be composed of points, 
for these either must be separated by some dis- 
tance, or they must become merged in one, with- 
out extension. A line is that which has fixed in- 
termediate parts extending through all the dis- 
tance between any two points contained in it ; but, 
at the same time, a line has in itself a continued 
succession of points which remain in their places ; 
and hence it becomes like a list or catalogue of 
points bound together by perfect continuity, every 
intermediate part of the line being as stable as the 
rest. Nevertheless, the nature of continuity is a 

a2 



10 

mystery which can never be understood. But it 
is evident that the extension or production of a 
line must begin from a fixed point, which is first 
found by the power of Jupiter. The first form of 
continuity must be a straight line ; of which the 
definition is, that it has equal relations on both sides. 
Supposing two positions (A 
and B) to be assumed equally ^^ B 

distant from the fixed point \\ NsjC ^'/* 
(C) from whence the line be- \ \ 

gins ; and, if the line, as it is \ ] J) / 

drawn out, preserves always 
an equality in its two distances 
(AD and BD) from these two lateral points 
(A and B) then it will be a straight line, and may 
be prolonged as such to any extent. But this 
also gives the first and simplest idea of an abstract 
rule prolonged and carried into successive in- 
stances; because the two lateral distances, though 
continually increasing and changing as the line ex- 
tends, retain always the same proportion to each 
other ; but their proportion to the extent of the 
line is continually diminishing. Among the 
twelve chief heathen divinities, continuity, or ab- 



11 

stract rule, was represented by Apollo, whose 
chariot was drawn through the sky by celestial 
steeds ; for the mystery of continuity relates to 
the yoking of different points together ; and the 
point which advances is bound to that which is 
thrown behind, in the same manner that the horses 
are bound to the car which they draw. This 
power may also be considered in another point of 
view, for each side of a line has relation to any 
point situated out of the line, and lying on that 
side. This generates the first idea of relation 
spreading into breadth, like that of a plane, be- 
cause the relations of the extraneous point (B) to 
the whole of the points situated in the line, are 
blended into one Maya, or ideal extension having 
breadth. Maya is the word used by the Indians 
to signify mental creation or illusion. This Maya 
was probably signified in Latona, whose character- 
istic, according to Hesiod, was smoothness and 
placidity. The lateral extension changes its 
boundaries as the line extends, and exhibits form 
in a state of transition through successive instances 
all belonging to the same rule. This elementary 
power must have belonged, among the tribes of 



12 

Israel, to the sons of Levi. Continuity is also 
the same as faithful tradition, which transmits 
what has been heard, without departing from the 
method which has been begun. Thus, the hard 
and accurate style used by the Egyptian carvers, 
in tracing the forms of animals, shews a rigid ad- 
herence to tradition ; and such is the strength and 
certainty which exists in the joints of the crocodile 
of the Nile. The name Phoebus, in Greek sig- 
nifying pure, applies well to the nature of abstract 
rule, which, in ideal generation, admits of no ex- 
ceptions ; for either the same principle is carried 
on throughout, or that rule is abandoned, and 
some other substituted in its place. 



13 



CHAP. IV. 

ON CONTRAST OR DIFFERENCE OF RULE. 

The fourth mathematical element is an angle 
which causes the line to assume a new direction, 
by which it comes to have a lateral relation to the 
previous part of the 
line (CF) ; and, there- 
fore, draws forth a 
maya as to each dif- D 
ferent point (C,D, E ? ) 
situated in that part. 
In each of thesemayas, 
the lateral distance 
(EG and DG and CG) increases according to a 
different rate, and exemplifies a different rule ; 
and the power of angle, therefore, becomes the 
same as the contrast of different rules or methods 
carried on at the same time, the reflected line (FG) 
being capable of prolongation any extent. Among 



E 






a 



~r>^\ . 



14 

the twelve chief heathen divinities, the power of 
angle was represented by Diana, the goddess of the 
moon, which sends back reflected light; and, among 
the tribes of Israel, the same power must have been 
represented by Judah, the tribe to which belonged 
superiority and dominion over all the rest. The 
characteristic of this elementary power is, that, 
without the aid of any extraneous point not con- 
tained in its own lines, it is capable of drawing 
forth an infinite number of mayas, or different 
forms of flowing extension within itself; because 
these are always produced between the two parts 
of the line ; but only the second part of the line is 
capable of prolongation, the prior part standing 
still, and affording fixed points, from whence the 
different mayas or rules are drawn forth to infini- 
tude. These pursue each other, from whence 
perhaps the idea of the chase attributed to Diana. 
But the flowing extension altogether, which is in- 
cluded in the form of the angle, was perhaps signi- 
fied in the character of Maia, and the Pleiades, 
the daughters of Atlas, who, being changed into 
stars, were made to shed a sweet and beneficial 
influence. Maia was called bountiful, and her 



15 

name signifies also a midwife or nurse, which ap- 
plies well to the origin and increase of subject ex- 
istence in the angle, one side of which is extending. 
It is evident, however, that one of the mayas which 
are drawn forth must be outermost and must out- 
strip all the rest. For this reason, the elementary 
power of Diana Corresponds with the nature of 
pride, and the wish to be highest, or to achieve 
the most. Such was the scriptural character of 
Lucifer, the son of the morning ; and, as will 
afterwards be shewn, such was likewise the mytho- 
logical character of Hercules, as belonging to the 
mode of composition which has most likeness to 
this elementary power. From possessing in itself 
the continuance of innumerable rules, which are 
all different, the elementary power of Diana be- 
comes the same as power over the different modes 
of action, to choose among them, or contrast them. 



16 



CHAP. V. 

ON DEFINITION, OR THE LIMITATION OF 
ABSTRACT RULES. 

The fifth elementary power is the continuation of 
the same line through successive angles, by which 
the unlimited continuation of the same rules or 
mayas is prevented. By a second angle there is 
intercepted an intermediate portion of line which 
can extend its own mayas no farther. The second 
angle, to shew the power of alternation which be- 
longs to it, must be supposed to be formed on the 
other side of the line ; and the succession of angles 
so alternating may be continued to infinitude. 
Among the twelve chief heathen divinities this 
power belonged to Vulcan, and is the same as the 
power of definition, because it relates to what is 



17 




situated on both sides of a line or boundary. 
Thus, in a series of alternating _ 
angles, any portion of line (EF) 
draws forth a maya of its own, 
in relation to the point (C) at 
the beginning of all the line ; 
and this maya, crossing any in- 
termediate portion of line (DE) 
is found on both sides of it, and 
so becomes like the power of be- 
ing present on both sides of a 
boundary, and so perceiving both what is contain- 
ed within it and also what is situated beyond it, 
and every successive portion of line draws forth 
its own peculiar may as or rules, but the whole 
series of them is comprehended in the general 
power of Vulcan. Among the twelve tribes of 
Israel this power must have belonged to Zabulon. 
By two or more angles the continuation of a 
straight line is also capable of returning to a 
former point in its own course, and so including a 
space or forming a terminated figure, which, in 
the simplest case, would be a triangle. The 



18 

power of Jupiter also marks off a space or figure, 
but only by relations alternating between distant 
points. The kind of boundary which belongs to 
Vulcan consists of line or continuous extension 
having everywhere a series of fixed points contain- 
ed in it. The series of alternating angles belong- 
ing to Vulcan is the form which appears in light- 
ning ; and he was also the god of fire, and the fa- 
bricator of thunderbolts. A second angle pre- 
vents the continuation of the flowing mayas which 
were generated in the first angle, and which de- 
pended upon the prolongation of the second part 
of the line, from whence the relation of Vulcan to 
halting or stopping, and shewing deficiency, which 
is the source of the buffoonery ascribed to him. A 
triangular figure encloses the flowing mayas alto- 
gether within certain boundaries ; and the same is 
the case with any other terminated figure, what- 
ever be its form or the number of its sides. The 
power of Vulcan is also the same as the end of 
line or of continuity. Without such limitation, 
there could be no particular continuity apart from 
other things; therefore, to Vulcan belongs the 



19 

power of separating those qualities which are con- 
tained in a definition from those qualities which 
are left out or denied by its terms, 



20 



CHAP VI. 

ON INDEFINITE VOLUME OR DIFFUSION. 

The sixth mathematical element is the power of 
rising through the difference of planes, from 
whence comes the generation of volume. If the 
power of Vulcan, or terminated figure, be assum- 
ed for the basis ; and if, from one of its angles, as 
a fixed point, a new line be made to rise, leaving 
the plane below, it will draw forth two new may as 
or surfaces, beginning from the two sides of the 
angle which is left beneath. These two surfaces 
follow the extension of the rising line, and, be- 
tween them, will be generated the first idea of 
bulk or volume, having the original terminated 
figure for its basis, and having one of the rising 
planes for its boundary, on each side ; but its in- 
crease altogether accompanies the prolongation of 
the rising line. Among the twelve chief heathen 



21 

divinities this power was represented by Neptune ; 
and, among the tribes of Israel, the same character 
must have belonged to Issachar. The birth of 
volume is like nature, or the first diffusion of the 
waters of chaos, which were subject material, and, 
at first, had nothing to prevent them from spread- 
ing indefinitely. Volume is only a secondary kind 
of maya, and differs from the first in its degree of 
affinity to lines, in which affinity it is altogether 
inferior to the first. The maya of Neptune is not 
produced under lines, but only under planes, 
which are themselves a kind of ideal creation; 
and volume is therefore a subject existence pro- 
duced under subject existence. According to the 
philosophy of the Indians, that which constitutes 
the nature of material substance is altogether an 
illusion, and has no real existence. In a plane, all 
the relations are clear and certain, because they re- 
fer to fixed points, situated in lines, and having an 
order or series ; but, in the maya of t Neptune 'or 
volume, the internal relations are of a different na- 
ture, because no fixed points can be found in the 
planes between which they are generated. Hence 



the character of Proteus, as a sort of illusive and 
unintelligible nature. Nevertheless, to the utmost 
height of the line of Neptune, there must be found 
a succession of fixed points, each of which has its 
own maya of volume in relation to the plane be- 
low ; but all these forms of volume are blended 
into one, which is continually increased as the line 
ascends. This power is the ideal source of me- 
lancholy, and obscuration from the intermediate 
parts of volume, as in the depths of the ocean ; 
but Neptune was also reckoned the god of intel- 
lect. A certain character of darkness and native 
sadness must be ascribed to all the marine powers, 
such as Oceanus, Tethys, Doris and her nymphs 
accustomed to twilight under the waves, and re- 
joicing in the obscurity of clouds and showers. 
Nevertheless there must always be a highest or 
outermost part of volume not blended with any of 
the mayas beneath ; and thus Neptune was con- 
ceived as generally driving his chariot, in day- 
light, over the surface of the brine, while the me- 
lancholy of the ocean was confined to the depths 
beneath. The power of Neptune is also the origin 



23 

of the form given to spires and pinnacles which 
are placed on the highest parts of buildings, and 
meant to appear as if rising into the sky, and ca- 
pable of unlimited ascent. 



24 



CHAP. VII. 

ON TERMINATED VOLUME. 

The seventh mathematical element is the power 
of separating the different forms of volume ; for, 
if the line of Neptune be made to form an angle, 
the next point in it will produce a maya of volume, 
which no longer entirely absorbs those beneath, 
but, having its apex situated in a new line, leaves 
the previous form of volume partially uncovered. 
This is like the appearance of dry land ; and every 
successive point, in the new line, becomes the apex 
of another form of volume, partially separated 
from the rest. Thus, by the second part of the 
line, there is produced a ridge of pyramidal mayas, 
having their forms partially separated from each 
other, and no longer capable of being lost in the 
general increase of volume. And, if the rising 
line be made to pass through other angles, it will 
only produce farther exemplifications of the same 

1 



25 

principle. But, if the line be made to descend 
and return to the original plane of Vulcan, and be 
fastened to some point in the lines which form its 
limits, then there will be produced a terminated 
solid, incapable of farther increase, and having, on 
all sides, planes which enclose the mayas of volume 
within. This power of solidity belonged to Vesta, 
among the twelve chief heathen divinities; and, 
among the tribes of Israel, the same character 
must have belonged to Dan. The line of Nep- 
tune, after forming an angle, which should be on 
the other side of the line, to shew its power of al- 
ternation, may be called the line of Vesta ; and in 
relation to the previous part of the same line, it 
draws forth a plane of its own, at the sides of which 
the two other planes belonging to each apex are 
joined. 

These are left by twos, as relating to a fixed point 
for their apex ; but the middle plane produced by 
the line if Vesta continues to extend, till another 
angle be formed at which that ridge terminates, 
and another begins. Farther, the sides of these 
ridges will not be uniform surfaces, but will consist 
of two different planes diagonally joined. The ge- 



26 

neration of this figure is regular and beautiful, but 
cannot easily be expressed by language ; nor could 
it well be represented by lines drawn on a plane. 

Another characteristic by which the power of 
Vesta is distinguished from that of Neptune is, that 
every point in the line of Vesta has two planes op- 
posite to it, in relation to each of which it produces 
a maya of volume ; so this power becomes the same 
as the coincidence of different mayas of volume in 
regard to the same point as the apex ; and this is an 
union which can never happen by the power of Nep- 
tune. The line of Ops or Vesta is the power of se- 
parating the forms of volume, and finding out new- 
examples of terminated solids or individual na- 
tures, the varieties of which it may explore to in- 
finitude. 

It is evident that solids, when complete in them- 
selves, and externally separated from others, are 
capable of relative motion, and of freely changing 
their places. The name of Rhea, in Greek ~Pux, 
signifying " flowing," may refer to the flowing of 
multitudes which are capable of freely changing 
their places ; as is exemplified in a shoal of fishes 
moving through the waters. The external rela- 



27 

tion of solids to each other also gives birth to a 
new kind of maya, alternating between them, and 
having relation to their figures and sizes. This 
may be called the maya of Faunus, a rural power 
belonging to the earth. 



2S 



CHAP. VIII. 

ON THE PROSPECTIVE RELATION OF SOLID 
FIGURE TO MOTION. 

The eighth mathematical element is the relation 
of the form of a solid to the various courses 
through which it might move ; and this is the 
source of the infinite extension of parallels ; be- 
cause whenever a solid begins to move, it must 
trace parallel lines, which are the sides of its course, 
if it continue to present the same front in moving. 
The idea of parallel lines situated in a plane is first 
found belonging to the power of Vulcan. But, 
when that which moves is a solid, the parallels which 
it draws are not situated in one plane, but in every 
plane; and the course which the solid traces is 
therefore also solid. But this power assumes an- 
other form; because, even when the solid is at 
rest, it has what may be called an anticipation, or 
prospective view, of all the solid courses through 



29 

which it might move; but, from being solid, they 
are blended together at their origin, and still con- 
tinue partially to retain their union as they diverge, 
altogether forming one maya, which is that of the 
courses through which the solid might move. 
Among the twelve chief heathen divinities this 
power was represented by Mars ; and among the 
tribes of Israel it must have belonged to Gad. In 
every solid, although the number of sides be limit- 
ed, there is an infinite multiplicity of aspects, to 
each of which belongs a solid course, in the form 
of parallels extending to infinitude ; but it is evi- 
dent that every course must have a different solid 
form, according to the aspect from whence it pro- 
ceeds. To the power of Mars belongs a sentiment 
of freedom, might, and power, extending far be- 
yond the local existence of the individual ; but this 
is a feeling which might also hypothetically belong 
to every particle among the sands of the sea. The 
union of the different courses must be most dense 
at first ; and, as they diverge and separate, the 
general maya, which they form, becomes rarer, 
like an aerial haze departing from the solid. 
The elementary power of Mars is the beginning 



30 

of a new series of powers all belonging to indivi- 
dual being; because all of them are produced 
from the nature of courses which originate from 
solidity, or the power of Vesta. The maya of 
Mars, however, is distinguished from any former 
maya by the characteristic of not being produced 
under lines or planes already extended ; for it is 
only prospective or potential. 



31 



CHAP IX. 

ON ACQUIRED MOTION. 

The ninth mathematical element is the generation 
of the parabola from the bending of parallel courses, 
since these cannot be prolonged without encounter- 
ing other solids ; as every solid has relation not 
merely to all the courses through which it might 
move, but also to all the opposition which it would 
meet with in them ; and from hence, in abstract 
idea, is the beginning of gravitation. For the sake 
of illustration, let it be considered that, if two pa- 
rallel lines be required to bend, and yet to retain 
the same distance between them, it is evident they 
must turn otherwise than by an angle, in which 
there will always be an oblique distance between 
the lines. Therefore it becomes necessary that 
the lines assume the form of two parallel curves, 
one of which, being the outer, must be the larger. 
For this purpose one of the lines must borrow and 



32 

gain from the other a continued increase of exten- 
sion ; but, if the outer line, at first, were to begin 
suddenly b) r borrowing some certain quantity, the 
curvature would begin suddenly, without any 
previous gradations between it and the original 
straightness of the lines, or their equality of pro- 
portion ; and if the borrowing were continued at 
the same rate, the two lines would bend together 
as the circumferences of a larger and smaller circle, 
and consequently would be finite. But this mode 
of beginning and continuing the curve is hypothe- 
tically impossible, because the first beginning of 
change of proportion between the straight lines 
must always have been infinitely remote, and in 
the least possible degree. 

The outer line, therefore, must have been gain- 
ing extension from the other, at a rate always in- 
creasing ; and the outer line is supposed in these 
two curves to be found borrowing according to 
the same proportion as exists in the decreasing 
series of numbers, or their approach to unity by 
continued diminution as in the series five, four, 
three, two, one ; and this is the kind of propor- 
tion which is expressed in the parabola. In this 



33 

mode of exhaustion, the unit which is taken away, 
at each step, becomess always a greater quan- 
tity in proportion to the number which remains to 
be exhausted. But if this principle be carried back 
towards the beginning of the curve, it would make 
the first change of proportion infinitely remote, and 
incapable of being found. Thus, if the outer line, 
in a certain portion of the course, had been gaining 
from the inner line one-seventeenth part, and had 
become to it as eighteen or sixteen ; and if, in the 
next preceding part, (in which the length of the 
outer and inner lines taken together had the same 
total amount), the outer line had been borrowing 
only one-eighteenth part, and had become to the 
inner line as nineteen to seventeen ; then, in the 
part still anterior to that, the outer line would have 
borrowed only one-nineteenth part from the inner, 
and would have become to it as twenty to eighteen. 
Thus the difference becomes always less ; and this 
progressive refinement or change of proportion is 
capable of being pursued for ever, and is express- 
ed in the proportions assumed by the parabola, 
when retraced towards its origin, which can never 

b2 



36 



CHAP. X. 

ON GROWING, OR THE ASSUMPTION OF 
EXTRANEOUS FORCES. 

The tenth mathematical element is the generation 
of the hyperbola from the assumption of an ex- 
traneous force which, being united or blended with 
those already operating, gives birth to a new form. 
If, at any part in the united courses of the para- 
bola, another straight course, or form of Mars, be 
applied, and made to take a share of curvature from 
the two courses which are already bending in the 
parabola, so as to bend along with them, and enable 
all the three to proceed together as parallels united 
in one curve ; in this case the newly-added course 
must communicate part of its straightness to the 
two former, and must receive from them part of 
their curvature. This produces an evolving form 
which is the hyperbola, and which is the original 
form of support, or nourishment, or vegetation, 



37 

which is from the assumption of extraneous forces. 
That which characterizes this elementary power is 
the sharing of the curvature among the courses 
which are united as parallels. Any quantity, con- 
tinually lessened by taking away the same part 
from the amount which remains, will give a series 
of proportionals in continued diminution ; and this 
is the rule or proportion which is expressed in the 
decreasing curvature of the hyperbola. Thus if, 
to the two parallel courses of the parabola, an- 
other single course be added and made to extend 
along with them, assuming a portion of their cur- 
vature, its share will be always one-third of the 
curvature which remains to be exhausted. There- 
fore, after each farther extension of the courses 
through another equal length, the quantity of cur- 
vature left will be only two-thirds of what it was 
before. It would first be as two-thirds of the ori- 
ginal quantity, then as two-ninths, and then as two 
twenty-sevenths, and so on in continued propor- 
tion. The hyperbola also in expanding repeats 
the same form of curvature in lengths which in- 
crease in continued proportion. Thus, if a certain 
form were found in a length which was as three ; 



38 

and if the same form were next found, on a larger 
scale, in a length which was to the former as nine 
to three ; then the same form would next be found, 
on a still larger scale, in a length which was, to 
the preceding, as twenty-seven to nine ; and so on 
in continued proportion. The hyperbola, there- 
fore, corresponds with the geometrical series of 
numbers. Among the twelve chief heathen divi- 
nities this power was represented by Ceres, the 
goddess of vegetation; and among the tribes of 
Israel the same character must have belonged to 
Naphthali. When the decrease of curvature is 
slow, the hyperbola evolves as a spiral ; but, in 
other instances, it takes place so rapidly that the 
curve forms no circumvolutions, but is capable of 
extending beside an infinite straight line placed be- 
side it as an asymptote. This is the form of the 
hyperbola found by that section of the cone. It may 
easily be shewn that, besides the parabola and hyper- 
bola, there can be no other simple and continuous 
forms of curves expressing change of quantity ; be- 
cause quantity can increase or diminish only in two 
continuous manners ; namely, first according to the 
sam e proportion which is expressed in an arithme- 



39 

tical series, and which belongs to the parabola ; 
and, secondly, according to the proportion which 
is expressed in a geometrical series, and which 
belongs to the hyperbola. To Ceres must be 
ascribed elasticity and expansion ; and the powers 
of recoil, which belong to elementary particles, al- 
ways operate according to a series of proportionals. 
The hyperbola is the last continuous line found in 
the mathematical deduction of forms ; and there 
is no end to its extension; for, if there remains 
any curvature at all in a line, it can never be ex- 
hausted, however long it may continue to be shared 
by another line extending along with it as a paral- 
lel, and adopting a form, in which their qualities 
unite. But the outer and inner lines or courses, 
in the hyperbola, are different curves, otherwise 
they could not be parallel ; and this is proved by 
the difference of the forms of parallel ellipses, 
traced within each other. The same hyperbola 
traced on a larger scale would diverge. 



40 



CHAR XI. 



ON COMPOSITION. 



The eleventh mathematical element is that from 
whence originates the power of composition ; be- 
cause, the eleventh element is not a continuous 
form generated according to one rule, but includes 
in itself the juncture of two forms extended ac- 
cording to different rules. If, at any part in the 
first hyperbola, another straight course, or power 
of Mars, be added, and made to proceed along 
with the first three united courses, as an addition- 
al power taking a share of the original curvature, 
it is evident that, after this addition^ the hyper- 
bola will not continue to extend according to the 
same rule, but will start off in a new direction, 
and diminish its curvature more rapidly than 
before ; because, after the juncture, the power 
which is sharing or diminishing the original cur- 
vature has become twice as much as before. A 



41 

second form of the hyperbola will therefore be ge- 
nerated, departing, at a crisis or juncture, from 
what would have been the continuation of the first 
curve. This shift or change of direction will be 
towards the outer side of the curve ; but the conca- 
vities of both the first and second parts will still be 
towards the same side. The share of curvature as- 
sumed by the first-added parallel was one-third ; 
but the first form of the curve being left, the pa^ 
rallel courses are continued into another form, in 
which the quantity of curvature assumed by the 
added courses becomes one-half; and, therefore, 
after each farther equal extension of the four pa- 
rallel courses together, the curvature left is only 
one-half of what it was before, and thus the quan- 
tity diminishes according to continued proportion 
in the second curve, as in the first, but according 
to a different rate. In this form are found two 
different qualities of curve, their partition being 
marked by a curve, angle, or juncture, at the place 
where the hyperbola branches off, and assumes a 
new direction. Among the twelve chief heathen 
divinities, this power was represented by Minerva ; 
and among the tribes of Israel, the same power 



42 

must have belonged to Joseph. From containing 
in itself the juncture of forms essentially different, 
this elementary power the origin of composition, 
which is the freedom of prolonging the same con- 
tinuity according to a rule different from that 
which has been begun. The ellipsis is an ex- 
ample of composition; because it is a figure not 
found by the continuation of a single curve, 
but composed of four similar parts joined to- 
gether in four inverted positions ; and only one- 
fourth part of it is continuous, being the same 
as the hyperbola, in abstract principle. By 
the power of Minerva, the prolongation of a line 
is enabled to change, at once, from any quali- 
ty of form to another ; and, in flowing lines, the 
junctures or crises cannot be easily detected. The 
elementary parts of form are capable of extension 
only according to one rule ; and the minutest parts 
of every line must be mathematical forms ; because 
a line cannot consist entirely of junctures or tran- 
sitions from one kind of extension to another : and 
if there be any portion, however minute, in which 
the mode of extension does not change, that por- 
tion must be a mathematical form which is part 



43 

either of a circle, a straight line, an angle, a pa- 
rabola, or an hyperbola ; since every rule in a 
plane must produce some one of these. To Mi- 
nerva, therefore, must be ascribed the power of 
elegance, or choice and freedom of transition from 
one principle to another. Considered as an ele- 
ment found by continued deduction, Minerva is a 
second form of Ceres or growing, and is produced 
from the assumption of a second extraneous force, 
after the power of the first extraneous force had 
been for some time exemplified. But the essence 
of this element is the juncture which it includes, 
and which necessarily implies that both the first 
and second curves must appear in the power of 
Minerva, when it is viewed as a single element, 
apart from others. 



44 



CHAP. XII. 

ON COMPREHENSION AND IMAGINATION, 

The twelfth and last element, found by continued 
deduction, is a general curve, comprehending a 
succession of different hyperbolic parts. For, if 
to the second hyperbola, which belongs to Miner- 
va, a third extraneous power or parallel course be 
added, it will only have the effect of producing a 
third hyperbola, which again starts off in a new 
direction, and exhibits a new quality of curve, con- 
stituting the third branch ; and, if farther addi- 
tional forces be successively added, they will only 
have the effect of producing a succession of hyper- 
bolic forms, changing their quality in continued 
gradation ; and, having the united courses of the 
original hyperbola continued through the whole of 
them ; but with the addition of one more parallel 
course added at the beginning of each successive 



45 

branch. And if the additions be made at such 
points as to have always equal lengths of course 
between them, then all the points or junctures, at 
the inner side, will be situated in an imaginary 
curve, which is also an evolving hyperbola, tend* 
ing in the same general direction as the particular 
branches, and having its convexity to the same 
side as theirs. This curve is not actually traced, 
but derives an ideal existence from the nature of 
the comprehending form. This elementary power 
was represented by Mercury, among the twelve 
chief heathen divinities ; and, among the tribes of 
Israel, the same character must have belonged to 
Benjamin. This is the last elementary power 
which is found by continued deduction ; since the 
successive assumptions of extraneous forces will 
produce no other form, although they be continued 
to infinitude. To express the relation of Mercury 
to number it was allegorically said that, while he 
was yet an infant, Juno had been induced to 
suckle him ; but, discovering that he was the son 
of Maia, she put him away ; and, the milk flow- 
ing from his lips through the celestial regions, 
produced the milky-way. The share of curvature 



46 

assumed in the third hyperbolic branch is three- 
fifths ; in the fourth branch, four-sixths ; in the 
fifth branch, five-sevenths ; and so on, the differ- 
ence of quality in each successive branch becoming 
always more refined. Mercury was called the in- 
terpreter, because of the power of an intermediate 
branch intervening between two others, and con- 
necting them, although they are entirely different 
in quality. To Mercury may also be ascribed the 
form of the syllogism; in which the minor, or 
second proposition, connects the first proposition 
with the conclusion. Three successive branches 
of the hyperbola are a mathematical syllogism, in 
which the original parallel courses are carried 
through an intermediate curve, and transferred 
into a third curve, of which the quality is deduci- 
ble from the first. Thus, by reasoning, the force 
of an acknowledged proposition is carried into 
other applications. But the characteristic of this 
elementary power is the comprehension of particu- 
lar forms in a general one ; which may be called 
the Saturnian hyperbola ; because of its fulness of 
parts, and because of the imputed character of 
time, which is poetically said to swallow up all 



47 

particular forms of existence. The Saturnian hy- 
perbola, besides occupying a real place by the suc- 
cessive forms contained in it, has also reference to 
a series of points situated in a totally different 
curve ; but the relations between these points 
would be direct, and like straight lines connecting 
them, and, therefore, would not in reality trace 
the curve ; which, nevertheless, is traced by the 
power of imagination ; and, therefore, to Mercury 
belongs a new kind of maya or ideal creation, dif- 
ferent from any other ; because it is not produced 
by the direct relations between points existing in 
any form really extended. Thus, the Saturnian 
hyperbola teaches the mind to create relations 
which are not there ; and this is the nature of 
fancy or conception. The twelfth element, there- 
fore, has two different forms, namely, that which 
has real extension and place, and that which is 
only imagined or conceived. Some of the ancients 
ascribed to Mercury the invention of letters, and 
said that he had learnt them from seeing: cranes 
tracing figures in the air by their flight ; and it 
is evident that these forms must have been dis- 



48 

covered by conceiving the relation of the crane to 
places in which it no longer was, and which were 
beyond its real or local extension. 



PART II. 

ON THE MODES OF COMPOSITION. 



CHAP. I. 



ON THE POWERS OF GRADATION, OR THE COM- 
PARISON OF DIFFERENT FINITE SERIES. 

Orion or Bacchus. 

The last of the simple elements is the Saturnian 
hyperbola, which comprehends a series of dif- 
ferent branches all included in one general curve. 
Beyond this no other elementary power is found. 
But, by resorting to composition, a new series of 
infinite forms is discovered. Composition must 
begin from the application of the power of Juno, 
which is the first. 

If the power of Juno, or alternation, be applied 
to the Saturnian hyperbola, it can have no other 



50 OIIION OR BACCHUS. 

effect than to make it alternate, that is to say, to 
make each branch cross itself, and so change its 
direction to the other side. But this change of 
direction is balanced or corrected by the succeed- 
ing branch also crossing itself and again extend- 
ing in the opposite direction ; so that the whole 
series, instead of bending as a general curve, is 
forced to stretch forward and assume a form some- 
what like that of alternating angles. To this 
form, however, there is a limit; because the 
breadth or thickness of each branch is increased 
by another parallel course added to it ; and the 
thickness of a single branch must at last become 
greater than will permit of its crossing. Thus, 
after a certain number of branches have alternated, 
the transverse progression comes to an end, and a 
limited series is found. If the series of branches 
be farther prolonged, they must resume the ori- 
ginal form of the Saturnian hyperbola. 

The power of Juno, when employed alone in 
composition, could not produce any infinite pro- 
gression ; and the second power, which is Jupiter, 
must therefore be applied. Jupiter is the repeti- 
tion of an identical form on a smaller scale, as in 
concentric circles. If this power be applied to the 
succession of hyperbolic branches, it will have the 
effect of transposing the remainder of it into a 
smaller scale, in which the same deduction is con- 
tinued. For it is evident that, on whatever scale 



ORION OR BACCHUS. 51 

an hyperbolic form is produced, it remains the 
same as to quality, and has still the same proper- 
ties as a particular curve. 

Suppose the transposition to be into a scale 
one-third of the former, then the breadth of a 
single branch will have become only one-third of 
what it was before ; but, if its length be drawn 
out to the same extent as formerly (as it must be 
to allow the investigation to proceed), then there 
will be a renewed freedom of making each branch 
cross and the series resume the transverse mode of 
progression; because the thickness of a single 
branch has now become less in proportion to its 
length ; and the thickness will be more slowly in- 
creased than formerly, because the same change of 
scale must be applied to the parallel courses which 
are added to the successive branches, and by which 
their thickness is increased. Therefore the new 
transverse progression will admit of a greater num- 
ber of branches than before, but it will also come 
to an end, and will constitute another finite series. 
After which, a new transposition into a scale of 
one-third must be again resorted to ; and, the 
length of a single branch, being drawn out to the 
same length as before, there will be a renewed 
freedom of transverse progression, which will also 
come to an end, and constitute another finite series 
still more extensive than the second. After which 
another transposition must be again resorted to ; 



52 OBION OR BACCHUS. 

and so on to infinitude. In each new transverse 
progression the number of branches will be greater 
in continued proportion ; because the rate of the 
increase made to the thickness of a single branch 
is changed after each transposition, and is diminish- 
ed in continued proportion ; so that the number 
of steps becomes greater, and the exhaustion of 
the whole series slower. 

The crossing of each branch within itself is the 
cause of measuring and intersecting the curve by 
other parallel lines going through it. These are 
the prolongation of the same curve, but being a 
different part, their form is different from that of 
the lines which they cross. The multiplicity of 
forms in the intersection is increased in each new 
branch by another added parallel. This mode of 
composition is the same as internal sensation or 
the consciousness of form ; and among the an- 
cients it was signified in the character of Bacchus ; 
because the effect of the vine is to produce an in- 
creased power of mental sensation, which is the 
same as the knowledge of form. In another point 
of view, this mode of composition was probably 
signified in Orion, the hero of the chase, or the 
power of exhausting intermediate distances. But 
this power, in effect, becomes the same as the com- 
parison of different modes of gradation, or the 
sense of beauty in finite portions of continuity ; 
because the transverse mode of progression is the 



ORION OR BACCHUS. 53 

same as the sense of continuity which binds con- 
trary points together and blends them into one 
form ; as each branch, in crossing, returns to the 
greater curvature in the preceding part of the 
branch, but is, at the same time, diminishing its 
curvature ; and this produces the blending of con- 
trary forms and directions, which resembles the 
binding of contrary points together in the con- 
tinuity of a line. Therefore Bacchus, like 
Apollo, was represented as drawn in a car. Con- 
tinuity is also the belt of Orion. But, as every 
transverse progression is finite, Bacchus can only 
be the same finite continuity ; or, rather, this 
mode of composition is like that comparison of in- 
dividual forms which belongs to the elementary 
power of Vesta, which draws forth successive ridges. 
It is therefore said in the Scripture, that the tribe 
of Dan should become as a judge among the rest. 
The comparison of different kinds of gradation pro- 
bably constituted also the characters of the Graces, 
Aglaia, Euphrosyne, and Thalia ; who, by some 
authorities, were called the daughters of Bacchus, 
by others of Jupiter and Venus. These minute 
goddesses were judges of refinement, and had the 
power of reproving coarseness or inelegance. The 
exhaustion of successive finite series, which in- 
crease in continued proportion, also extends to a 
sense of the unlimited ; because, although each 
particular series comes to an end, it is evident, 



54 ORION OR BACCHUS. 

from comparing those series which have been al- 
ready completed, that the number of steps in the 
future ranges to be found must pass beyond the 
limits of all calculation. This is probably ex- 
pressed in the Scripture in Jacob's ladder. Tak- 
ing a stone (that is Vesta) for his pillow, he fell 
asleep, and saw a scale extending up to heaven. 
Some poets, such as Pope, have applied the same 
conception to the gradations of being in the uni- 
verse, likening it to a great chain of which no link 
or intermediate degree could be wanted. This 
notion, however, is in a great measure erroneous, 
since the order of the universe probably depends 
more upon differences of kind than of degree, 
which latter is left open for progressive change, 
except among the brutes. 

To this mode of composition may also be re- 
ferred lyrical poetry, in which there is the com- 
pletion of limited forms or stanzas. This charac- 
ter belonged to the people of Mytelene, among 
whom lyrical poetry received its origin from 
Alcaeus. The transition into another series was 
also expressed in the action of Sappho, who leapt 
from the Leucadian rock, to escape from the bond- 
age of her passion. Theophrastus, a native of the 
same island, satirized the improprieties of man- 
ners, or the want of refinement and taste. But, 
mythologically, the exhaustion of a particular se- 
ries was expressed in the fable of Meleager the 



ORION OR BACCHUS. 



55 



iEtolian, the duration of whose life depended on 
the consumption of a brand snatched from the 
hearth. The ancient Romans, whose character 
corresponded with this mode of composition, 
thought that the highest glory and virtue was in 
the progressive enlargement of their prospects, or 
increase of range, as exemplified in their conquests ; 
for they were not contented to have their views 
terminated by any impassable barrier. These 
characteristics led them to approve of suicide. 
Among the modern inhabitants of Italy, the same 
taste appears in their love of the fine arts which, in 
regard to sensation, are essentially a trial of the 
various modes of gradation, or a comparison of 
different limited series. To the power of Orion 
belongs also emulation, which will not endure to be 
exceeded or surpassed. The character of the an- 
cient Romans is accounted to have expired when 
there was placed over their heads an emperor, to 
whom all the other officers of state were virtually 
accountable and subordinate ; for then the glory 
of their separate duties and powers appeared to be 
lost. Such is the nature of this mode of composi- 
tion as to republicanism ; but in the fine arts, 
where it is more useful, and can be more happily 
exercised, it tends to the multiplication of different 
departments, each having a manner and prospect 
of its own. Thus, republican virtue is converted 
into what is called vertu, or the sense of beauty. 



56 ORION OR BACCHUS. 

This mode of composition has also a certain re- 
lation to the nature of the parabola, which must 
always reach a termination, and which, in ap- 
proaching to that termination, traces the inter- 
mediate gradations of form according to the na- 
ture of a decreasing arithmetical range, in which 
the same quantity is taken away at each step. 
Now, in the exhaustion of any limited series, the 
same form is virtually expressed by the continual- 
ly changing proportion of a single step to the 
quantity which remains to be exhausted. There- 
fore, Orion is virtually the same as the power of 
searching out the different forms of the parabola, 
always passing from a more rapid form, in which 
the gradations are coarser, to a slower form in 
which the gradations are more refined, and, by 
their greater number, extend the range or com- 
pass of the form to which they belong. Suppos- 
ing the first series had thirty steps, the next 
transposition into a scale of one-third would begin 
a series of ninety ; which, by another transposi- 
tion, would pass into a series of two hundred and 
seventy, and so on ; each of these series having a 
form of the parabola which corresponds with it. 
One of the mythological representations of the 
finite series was Hebe, the cup-bearer of the gods ; 
and her character has relation either to the com- 
pletion or exhaustion of a finite quantity. This 
is also the character of Priapus, god of lakes, as 



ORION Oil BACCHUS. 57 

determinate quantities ; and, in another point of 
view, of Pomona, goddess of orchards, because 
the fruits are finite quantities or completions. 
From each branch crossing within itself, and 
so changing its direction, this mode of composi- 
tion is also like a search into the differences of 
direction ; and, therefore, it has a certain relation 
to the elementary power of Mars ; from whence 
one of his names, Gradivus. This, however, is 
but a partial resemblance, and must always come 
to an end ; since the changes of direction in each 
series are finite in their number ; although each 
new transposition produces a more extended range. 
Therefore, having only a partial resemblance to 
Mars, this mode of composition has most relation 
to the power of Vesta; and the character of the 
inhabitants of Italy is that which comes nearest to 
it. The fable of Saturn hiding himself in Latium, 
signified that the form of the Saturnian hyperbola 
is lost or hidden in the transverse mode of pro- 
gression, which is the same as Latium. This is 
the dominion of the Pope, whose character and 
place virtually coincides with that of Orion. 
Among the Homeric heroes, in the Grecian troops, 
the same power was perhaps represented by 
Diomed, whose native country was ^Etolia, the 
birth-place of Meleager. In the Trojan battles, 
the son of Tydeus was distinguished for his auda- 

c2 



58 ORION OR BACCHUS. 

city, which, like that of Orion, passed beyond all 
limits, and prompted him to lift his spear even 
against divinities. The same refusal to acknow- 
ledge any superior power was signified also in the 
fable of the giants, sons of the earth, or pow r ers of 
Vesta, who, wishing to scale the heavens, placed 
Pelion on Ossa, and, upon these, piled other 
mountains. 

The kings of Latium were descended from 
Faunus, a rustic power attached to the earth, or, 
in other words, belonging to Vesta. To the 
Fauns, however, there was ascribed a complete 
human figure, with the addition of small horns in 
the forehead, and a short tail at the termination 
of the spine. The Fauns may have had relation 
to the sense of form, which corresponds with a 
single branch crossing within itself and producing 
the intersections of parallel curves. Something 
like this is exhibited in the patterns of windows 
used in Gothic architecture, where parallel forms 
of the hyperbola are crossed by others, which, 
however, are generally exactly similar, and are 
not farther parts of the same curves, as must be 
the case when a single branch crosses by changing 
its direction. Closely connected with the sense of 
form is the love of the grotesque, which may also 
be ascribed to the Fauns, and which is not incapa- 
ble of being mingled with the sublime. These 



ORION OR BACCHUS. 



59 



forms, along with the Satyrs, accompanied Bacchus, 
a divinity ever young, and only reaching the 
end of any series to pass into another more exten- 
sive. But the chief characteristic of this power is 
the comparison of the different kinds of gradation ; 
and, from this comparison, the sense of unlimited 
range is generated. 



60 



CHAP. II. 

ON THE POWER OF SEARCHING OUT THE DIF- 
FERENT FORMS OF THE SATURNIAN HYPER- 
BOLA. 

Hercules and the Muses. 

The exhaustion of successive finite series, found 
by transposition into a smaller scale, may also be 
considered in another point of view, in which it 
becomes the power of searching out the different 
forms of the Saturnian hyperbola ; for, at the 
end of each transverse progression, if the same 
series of branches be continued without crossing, 
they will give a new form of the Saturnian hyper- 
bola, that is, a new and different example of the 
comprehending curve. It is evident that the ra- 
pidity of any single form of that curve depends 
upon the number of branches or accelerations it 
contains within a certain extent of course ; or, in 
other words, it depends upon the shortness of a 
single branch. Now, at each transposition into a 
smaller scale, the length of a single branch being 
virtually increased, it follows that, after the trans- 



HERCULES AND THE MUSES. 61 

verse progression has been completed, the farther 
continuation of the same series of branches, with- 
out crossing, must give birth to a new Saturnian 
form, slower than the preceding. This form, 
however, must not be considered as capable of 
infinite prolongation, but only as a short specimen, 
which is tried before the series of branches is again 
transposed into a smaller scale. The comparison 
of these successive specimens is the power belong- 
ing to the second mode of composition ; for the 
power of Bacchus does not necessarily imply the 
continuation of a series of branches after they have 
ceased to cross. Among the ancients, this second 
mode of composition was represented by Hercules, 
the god of achievement and acquisition ; and those 
persons who acquired wealth were said to have 
been befriended by Hercules ; because all the Sa- 
turnian forms are forms of comprehension. Each 
transverse progression terminates in the discovery 
of a new Saturnian form, and corresponds with 
the spirit of investigation and adventure. Thus 
Hercules, in his wanderings, pierced into the 
stupendous scenes of rocks in the Hyperborean 
regions. Contemplating the wild shapes of ca- 
verns which suddenly divided light from darkness, 
he heard the cries of the wolf or eagle, and saw 
the distant mountains covered with snow. Pur- 
suing his way through the labyrinth of rugged 
cliffs, he saw, beneath him, the concave edges of 



62 HERCULES AND THE MUSES. 

cliffs, as it were set with teeth, while at every step 
his powers of scrutiny were sharpened, and the 
keenness of his vision increased. Descending to 
a milder climate, he found the Pierian maids al- 
ways ready to welcome their associate, and to hail 
him by the name of Hercules Musagetes, or the 
leader of the muses. The ancients were accustom- 
ed to build temples to Hercules and the muses, as 
powers associated and joined together. The rea- 
son of this seems to be, that the muses are the 
powers of inventing and conceiving ; and imagina- 
tion, or the power of comprehension, which be- 
longs to Saturn, must also be attributed to them, 
who are rather the varied powers of comprehen- 
sion, or its different modes in the different arts 
and sciences to which fancy and invention are 
applied. Hence their various attributes of music, 
painting, lyrical poetry, tragedy and comedy, and 
the rest. But, in English, to muse signifies to 
fancy or conceive. 

This mode of composition, although its practical 
power is to search out the differences of Saturnian 
forms, has most relation to the elementary power 
of Diana or contrast, because the differences which 
it examines are those of forms of the same kind, 
namely, forms of Mercury. Now contrast is the 
comparison not of forms without similitude or rela- 
tion, but rather of forms which are allied by prin- 
ciple, so as to induce comparison of their courses. 



HERCULES AND THE MUSES. 63 

The most perfect contrast is in forms which are 
different specimens belonging to the same general 
rule ; because, between these, there is always a re- 
lation which bends them together and forces 
comparison, while they diverge and pursue dis- 
similar courses. Contrast, therefore, may be 
ascribed to Hercules and the muses ; and this 
mode of composition has most resemblance to the 
elementary power of Diana, and has relation to 
wealth and the aggregation of powers. To the 
same class with the muses may be referred Mor- 
pheus, a deity who had in his power all the diver- 
sity of dreams, or all the forms of the Saturnian 
hyperbola. 

The kind of mental feeling which belongs to this 
mode of composition may easily be known by its 
characteristics; for in every thing it desires the 
result. It also finds pleasure in the knowledge of 
particular examples of every power ; from whence, 
perhaps, the character of Eros, or the celestial 
Cupid, as the power of fixing on particular speci- 
mens. But from the love of contrast springs also 
allegory, which is some abstract truth presented 
as if disguised or hidden in a particular form or 
conception. Spenser is perhaps the poet near- 
est to English genius. Among the ancient poets, 
the most disposition to professed allegory is shewn 
in Anacreon, whom Love, in the form of a child, 



64 HERCULES AND THE MUSES. 

frequently calls forth to battle, or invites to run a 
race with him. 

To Hercules and the muses must be ascribed 
freedom and generosity of mind, and productive 
power ; because every particular specimen sug- 
gests the possibility of others. This characteristic 
leads again to the power of achievement and ac- 
quisition. It has been said that, in England, the 
power of science is always seen in its result ; and 
the character of the English nation is that which 
comes nearest to this mode of composition ; both 
as to achievement and the search after results, 
shewn in the doctrines of the chief English philoso- 
pher, Lord Bacon, and also as to the national 
disposition to acknowledge the freedom of indivi- 
duals, or to suffer each to pursue apart his own 
course, as one more added to the number of dis- 
similar modes of conception, which this power 
must always wish to see multiplied. The succes- 
sive transpositions into smaller scales have relation 
to the sense of the picturesque, from the boldness 
and suddenness of the results ; corresponding with 
which is the spirit of adventure advancing precipi- 
tately, while yet unable to guess the conclusion. 
The picturesque, in another point of view, seems 
to result from the composition of forms which are 
themselves composite ; in the same manner that 
the power of Hercules searches out different Sa- 



HERCULES AND THE MUSES. 65 

turnian curves, which are themselves forms of 
comprehension, including hyperbolic parts. Thus 
the picturesque may be produced from the rugged 
compilation of cliffs which exemplify different kinds 
of ruggedness. 

To the elementary power of Mercury belongs 

conviction. Now, this mode of composition being 

the means of searching out the various forms of 

the Saturnian hyperbola, is the same as the power 

of comparing the various modes of conviction, or 

of weighing and considering arguments together, 

Minos, a form of this power, was appointed to be 

one of the infernal judges. Ariadne, the power of 

continuous deduction, the daughter of Minos, gave 

to Theseus the thread or principle by which he 

was guided through that maze which resembled 

the perplexities and difficulties to be encountered 

in reconciling the different aspects assumed by the 

same truth ; for all the different Saturnian forms 

discovered are examples of the same curve. To 

Hercules or Minos, therefore, may also be ascribed 

the comparison of arguments, as a judge listens to 

what is said on both sides, till he finds out the 

truth, through the disguises of different statements. 

Diana was known under the name of Dictynna, or 

the Cretan, as if to signify the application of this 

character to her. Others said that the original 

Dictynna was a nymph beloved by Minos, and 

celebrated for having first invented hunting nets, 



66 HERCULES AND THE MUSES. 

and using them on Dicte, a mountain in Crete. 
This fable corresponds with the power of acquisi- 
tion. The voracity ascribed to Hercules was often 
a source of ridicule ; but in the Muses it would 
appear as the desire to be always acquiring or 
comprehending something farther. Thus the la- 
borious Aristotle, having explored all the different 
forms of syllogism, turned his powers of research 
into various other departments of science ; or thus 
St Paul would never have an end of adding one 
more to the number of those he had converted to 
the Christian faith. Among the apostles, the 
place of Minos or Diana seems at first to have be- 
longed to Judas Iscariot ; but afterwards the 
same character was shewn in a totally different 
point of view in St Paul, who, of all the sacred 
writers, shewed the most inclination for the exer- 
cise of reason, or for that opposition of arguments 
which belongs to Minos, the comparer of evi- 
dence. 



67 



CHAP. III. 

ON THE APPLICATION OF STRAIGHT LINES TO 
THE SERIES OF HYPERBOLIC BRANCHES. 

Pan or JEsculapms. 

The third mode of composition is from the appli- 
cation of the third power, which is that of Apollo, 
or straight lines, to the forms of the hyperbolic 
branches. The straight lines, proceeding from the 
beginning of the whole as from a centre, diverge 
and become the power of marking off successive 
portions of any particular curve, by which it is 
shewn to have one nature throughout, or to con- 
tinue producing similar forms. Or, when applied 
to successive branches, the straight lines are still 
capable of finding their harmony and relation of 
these branches, and of giving so much for so much, 
in two forms measured at once. Hence the balance 
may be ascribed to Pan, although it belongs truly 
to Pollux or Eridanus, as will afterwards be shewn. 
This mode of composition is like the concurrence 
of abstract rule with nature, to help it, support it, 
and bear it out; and it corresponds with the 



68 PAN OR ^SCULAPIUS. 

medica, or strengthening power of iEsculapius; 
for it sooths and flatters nature by going along 
with it. This mode of composition contains no 
power of creative progression ; but the straight 
lines, diverging from and moving round a centre, 
can only continue to measure or try those curves 
along which they spread. This mode of com- 
position was expressed in the character of Pan, 
the inspirer of alarms ; but he was also a wood- 
land power, loving to sit tranquilly amidst the 
forest recesses, and to diffuse the sound of his 
reeds through the trembling air ; and his pastoral 
attribute is another characteristic of the power of 
superintendence proceeding from one centre. This 
mode of composition has most relation to the ele- 
mentary principle of Jupiter, or revolution, and 
the completion of cycles. But since the parallel 
courses which extend together in any hyperbolic 
form must always be different curves, it follows 
that a straight line moving over them must al- 
ways measure different forms at the same time ; 
and must give or mark off corresponding portions 
in the curves which bend together as parallels. 
The characteristics of this mode of composition 
seem to belong to the Scotch nation, or more par- 
ticularly to the Scotch Highlanders. The Low- 
landers of Scotland may perhaps rather be taken 
for forms of Pluto, a power which will after- 
wards be explained, and which was also mythologi- 



PAN OR JESCULAPIUS. 69 

cally exemplified in Epirnetheus, deriving know- 
ledge from events after they were past, but never 
able to foresee them. But the character of Pro- 
metheus, who was distinguished for wisdom or 
foresight, in some respects corresponded with 
iEsculapius, as the power of animating and sup- 
porting. iEsculapius was the son of the nymph 
Coronis, who was afterwards changed into a raven, 
the type of foresight and vaticination. To the 
same mode of composition with Pan may be also re- 
ferred the Pythian Apollo, whose oracles were ut- 
tered at Delphi, in Phocis. The name of one of 
the places in that region was Anemoreia, that is, 
" the rushing of blasts," which is like some of the 
names in the poems of Ossian. In the Trojan 
war, the Phocians were led by Schedios and Epis- 
trophos ; but this power was not represented by 
any distinguished character among the Grecian 
troops. Of all the modes of composition, this, 
which coincides with the elementary power of Ju- 
piter, is the coarsest and the most vulgar, but, at 
the same time, it has a certain beauty of its own, 
which arises from its universality and simplicity. 
To St Andrew may be ascribed the pastoral flute ; 
and the essential characteristic of Pan is the incli- 
nation to superintend and interfere. Thus the 
Delphic oracle, by its responses, found means to 
direct the affairs not only of Greece, but of many 
surrounding countries. But Pan is likewise the 



70 PAN OR .ESCULAPIUS. 

power of all taken together ; and a system of mu- 
tual control and interference or usurpation, if it 
were perfectly equal, in a large community, might 
he compatible with the simplicity and kindness of 
the golden age, or the good-nature of mankind 
exercised in prompting and admonishing each 
other. This is the beautiful ideal of democracy. 
But it would preclude all difference of manners or 
habits ; for the characteristic of Pan is the inca- 
pability of progressive change ; and this is the 
true meaning of his form being partly similar to 
that of beasts. The conviction, that the same 
things which now are must continue for ever, con- 
stituted the sunshine of the golden age, and was 
exemplified in the unchangeable character of shep- 
herds engaged in superintending their flocks, or in 
sounding their vocal reeds. But Pan was also 
much reverenced by the ancients for the wonders 
he accomplished by inspiring panics in war, al- 
though the exercise of this power was generally of 
short duration. To this class may perhaps be re- 
ferred Napoleon Buonaparte, who is now for ever 
laid asleep in the island of St Helena. It was the 
opinion of Napoleon, as of many others, that every 
thing in the world ought to depend upon the mu- 
tual consent and concurrence of numbers. This, 
at first, has the appearance of being liberal and 
just, but it would soon preclude all the differences 
of manners, and make an end of individual free- 



PAN OR /ESCULAPIUS. 71 

clom; since the multitude leagued together will 
always be inclined to persecute and insult those 
who refuse to combine with them. The power of 
universal consent would knit mankind [together in 
a world of vulgar falsehood of their own establish- 
ing, and would keep them, like tribes of savages, 
ignorant of every thing beyond. There can be 
no difference of manners or opinions among the 
brute creation ; but, in proportion as men rise in 
the scale of intellectual life, it becomes more ne- 
cessary that individuals should have the liberty of 
keeping their motives and feelings apart from those 
of others. 



72 



CHAP. IV. 

9 

ON THE RETROGRESSIONS OF THE HYPERBOLA. 

SECT. I. 

ON THE INFINITUDE OF A SINGLE CURVE. 

Pelops or Proserpina. 

The fourth mode of composition is from the ap- 
plication of the elementary power of Diana, or 
angle, to the hyperbola. This is the means of 
shewing that the curve is infinite, in which ever 
direction it is traced ; or, in other words, that it 
has no real centre, or that its course may be trac- 
ed back for ever into the depths of space. For if 
successive portions of straight line be placed along 
the inside of the curve, touching it, and always 
joined to each other at the same angle, it fol- 



PELOPS OR PROSERPINA. 73 

lows, from the nature of the curve, that these por- 
tions of line will be a series of proportionals di- 
minishing towards the origin of the curve ; because 
they are joined together alw r ays at the same angle, 
and because the successive angles which they form 
all touch the inside of the hyperbola, and their 
series participates of its quality, which is founded 
on continued proportion. Now, when the origin 
of the hyperbola is arrived at, and when the curve 
extends no farther in that direction, as a guide, 
then the series of angles may still be continued, 
according to that proportion which is already be- 
gun, and they will continue to trace back the curve 
for ever, as it retires by diminution into the depths 
of space, and disappears from the reach of the 
senses. But the parallel courses of the original 
hyperbola must also be supposed to be prolonged 
in the retrograde direction, and to follow the series 
of angles. The breadth of the parallel courses 
w r ould at last become an obstacle to their farther 
retrogression ; but these also, by successive falls or 
contractions, greater in continued proportion, may 
be supposed to approach the innermost line, and 
so to permit a farther extension of the curve by a 
number of angles greater in continued proportion, 
after each fall. These will draw out successive 
forms of the same curve, always more extended ; 
these portions being marked off by the successive 

D 



74 PELOPS OR PROSERPINA. 

falls. Supposing the first portion had ten angles, 
and the next portion had twenty, then the second 
portion would contain the first form twice ; the 
third portion would have forty angles, and would 
contain the second form twice, and so on to in- 
finitude; each successive portion being a more 
extended form or reach of the innermost curve. 
But since, after each fall or contraction, the breadth 
of the outer parallel courses is made less, their forms 
must become different, while the innermost curve 
touching the angles continues always the same. 
Therefore, after each fall or contraction, the outer 
courses of the hyperbola are made to produce new 
hyperbolic forms, nearer to the innermost curve, 
to which they are always parallel ; and, the power 
of Proserpina, although apparently one curve, has 
an infinite series of different hyperbolic forms, all 
fitted to the outer side of the simple and continued 
curve which touches the angles. 

Farther, by this mode of composition, it may 
also be shewn that any hyperbola, although extend- 
ed in the evolving direction along an asymptote, 
will at last, in its retrogressive course, assume the 
form of an involute ; because the repetition of any 
angle, however obtuse, w T ill at last make the curve 
turn in. The characteristic of this power, neverthe- 
less, is that, being such as to withdraw from the reach 
of the senses, it can at last only be made a subject of 



PELOPS OR PROSERPINA. 75 

cogitation ; for the remainder of the form assum- 
ed by the hyperbola in its retrogression can never 
be seen, but can only be ruminated upon in silence. 
This is the kind of refinement and beauty which 
belongs to Proserpina, the queen of the shades, and 
which has the characteristic of purity and firm- 
ness in belonging all to one curve. But this curve 
may also escape from its perfect uniformity, and 
become susceptible of change of direction, by con- 
tinuing always to cross itself in the same direction 
towards the inner side of the curve. In this case 
the general course of the curve would continue the 
same as before, and every second portion of it 
would return to the original course of the simple 
form, but every intermediate portion being turned 
in the opposite direction, so as to have its concave 
side outwards, would give to the power of Proser- 
pina a sense of continually changing direction. 
Each crossing may be supposed to take place 
after an equal number of angles, and the number 
of crossings, after each fall, would therefore in- 
crease in continued proportion. Such may be 
imagined the meanders of the river Phlegethon. 
The successive intersections of the curve would 
also be connected with a consciousness of in- 
creasing refinement in the successive parts of the 
same form ; or the continued reproduction of the 
same form on a smaller scale by the same curve 
in its retrogressive course. 



76 PELOPS OR PROSERPINA. 

This mode of composition has most relation to 
the elementary power of Venus ; because the para- 
bola is retraced towards its origin, which is in- 
finitely remote, and can never be found. The 
power of Proserpina is similar to this ; but it re- 
tires within itself, and traces back the form of the 
hyperbola for ever to its origin, by means of 
cogitation or deep internal feeling. Thus Phle- 
gethon continued to restrain its sound the more 
in proportion as it descended into a deeper shade, 
till it was lost in silence. But, in another point 
of view to Proserpina may be ascribed the power 
of fashion, which excludes or refuses to acknow- 
ledge whatever is not found within a certain line, 
or manner, or quality ; in the same way that 
Proserpina refuses to acknowledge any form which 
is not found in tracing that one curve which has 
been begun. The refinements of social feeling 
may also be compared to the connexion of succes- 
sive forms, all having their places in one curve, 
and constituting a whole by the mutual corre- 
spondence of their quality or fashion which is con- 
formed to the same ruling principle, although the 
forms themselves must all have different places 
and functions in the series. Among the ancients, 
the country whose inhabitants seem to have had 
most relation to this power was Elis, where a gene- 
ral assembly was held of the Grecian nations at the 
Olympic games; and where, probably, at every 



PKLOPS OR PROSERPINA. 77 

celebration of the games, a sort of fashion or pre- 
vailing mode was communicated, to retain its in- 
fluence for the next five years. The inhabitants 
of Elis were exempted from the hardships of mi- 
litary service, and their region was veiwed as a 
privileged field, and as sacred to the refinement 
of all the nations. This was probably also the 
character of Pelops, whose shoulder was said to 
have been devoured by Ceres, and from whom 
the Grecian peninsula took its ancient name. In 
modern Europe, the same character is perhaps 
exemplified in the Prussians, as to the variety of 
particulars included in one fashion. To the same 
class may be referred the leopard, the most beau- 
tiful of the tiger species. This is perhaps also the 
true character of Ariadne, the companion of Bac- 
chus ; because of the increasing range and exten- 
sion of the retrogressive forms which are traced 
by the power of Proserpina ; in which respect it 
has a correspondence with the character of Bac- 
chus. But, in another point of view, the succes- 
sive falls in the breadth of the curve have relation 
to the condensation of substance, and its power of 
involution, which is expressed in the proboscis of 
the elephant, as condensation is in the tusks ; 
from whence perhaps the ivory shoulder of 
Pelops. 

In the Italic region the power of Proserpina 
seems to have belonged to the Etrurians, a nation 



78 PELOPS OR PROSERPINA. 

always distinguished for taste, and also for a cer- 
tain characteristic firmness and purity in these 
forms which they designed. The Etrurians are 
said by Herodotus to have been originally a colony 
from Lydia in Asia Minor. To the school of Pro- 
serpina may evidently be referred Michael Angelo, 
and Dante, the explorer of the infernal regions, 
who frequently endeavours to express what is be- 
yond the reach of the senses, as when, at the end 
of his poem, he endeavours to describe the visible 
appearance of the Trinity. 

This talent is of a nature exclusive, and often 
severe, but at the same time capable of uniting 
beauty with a stiff and peculiar manner. It is 
also compatible with recluseness ; because the re- 
cluse may be as a system of fashion to himself, by 
rejecting and excluding all that is not within a cer- 
tain line which he has chosen. To this mode of 
composition may perhaps be referred one of the 
modern English poets, namely, Wordsworth, who 
has the power of finding out much within any 
given limits. To the same power may perhaps be 
referred topography, which finds out all the re- 
sources and localities contained within a given re- 
gion. Thus, Dante investigated the different cir- 
cles of the shades. Galileo, applying the tele- 
scope to portions of the heavens, found in them 
clusters of minute stars, which had not before 
been discovered. The power of Proserpina, being 



PELOPS OR PROSERPINA. 79 

confined to one curve, has more relation to the dis- 
covery of different places than of different manners 
and kinds of extension. Nevertheless, though, for 
this reason, connected with stiffness and simplicity, 
it has a strong relation to the discovery of form ; 
which it augmented by the curve being capable of 
a sort of alternation, which changes the situation 
of particular parts, while its general course as an 
involute is preserved. 

Elis was also the place in which the muses were 
met by Thamyris, who offered to contend with 
them in song, and whom they overcame and de- 
prived of sight, because it was previously agreed 
that the party vanquished should be at the disposal 
of the victor. Tiresias was another blind prophet, 
celebrated for his wonderful knowledge, and for a 
fabled change of sex. That Proserpina was con- 
sidered by the ancients as one of the most refined 
powers, is shewn by the remark of Vitruvius, who 
says that, in building temples to Proserpina, Flo- 
ra, or Venus, the Corinthian order should be em- 
ployed as the most conformable to the delicate 
and feminine nature of these powers. There was 
also a restorative influence ascribed to the daughter 
of Ceres, and one of her titles was Proserpina the 
Salutary; perhaps from the power of order and 
regular distribution of parts within a certain space, 
which was before explained. 

In regard to feeling, Proserpina is the same as the 



80 PELOPS OR PROSERPINA. 

elementary power of Venus or acquired motion ; 
and this is also the character of Eros or love, in his 
most universal form. As in the physical world 
the gravitating power of a body relates to all the 
solids in the universe, and compares their respec- 
tive influences ; so the power of Proserpina is the 
same as the feeling of all things taken together ; 
from whence the character of the counsellor, Nes- 
tor of Pylos, who weighed different considerations 
with each other. Among the modern European 
poets this character seems to belong to Goethe, 
whose poetry exemplifies a sublime enthusiasm 
and sensibility in regard to the powers of the whole 
universe modifying each other. This is the same 
as the love of variety which liberates the mind, and 
enables it to feel an unconfined enthusiasm, in 
yielding to the powers of all external affections to- 
gether. This is also the character of Hesiod, 
whose theogony is imbued with a feeling of the 
influential powers of the whole universe, and of 
Love, the most beautiful of the divinities. Hence 
the image of jealousy spoken of in the scripture ; 
for this is the love of created being more than the 
love of God. Thus at the Olympic games, the 
ancients could scarcely perform religious rites with 
sufficient devotion for thinking of the vast assembly 
which was present. 



81 



SECT. II. 

ON A RETROGRESSIVE SERIES OF BRANCHES, OR 
THE POWER OF SEARCH INTO ANTIQUITIES. 

Pluto or Adonis. 

The application of a series of similar angles to the 
interior of a curve is capable of other powers bed- 
sides that which belongs to Proserpina, and gives 
birth to other modes of composition. Throughout 
the whole series, the angle itself must be supposed 
to remain the same, but the proportion of its two 
sides being capable of being changed, the series of 
angles may be employed to retrace different hy- 
perbolic curves. If, in the series of lines dimi- 
nishing according to continued proportion, the 
later or shorter line be increased to nearer the 
same length with the preceding, then the nature 
of the curve which is traced will be changed, 
and it will be transposed, in the retrograde direc- 
tion, into a slower form of the hyperbola ; which, 
after a certain number of angles, may again be 
changed into a slower, by another addition to the 
length of the later line. But the series of added 

d2 



82 PLUTO OR ADONIS, 

quantities must diminish in continued proportion, 
to prevent the later line from even becoming equal 
to the anterior ; for a series of additions, diminish- 
ing in continued proportion with sufficient rapidi- 
ty, will never increase the later line beyond a cer- 
tain extent. If this precaution be observed, the 
retrogressive series of branches will be infinite; 
and, if to each branch the same number of angles 
be allowed, the lengths of the branches will dimi- 
nish in continued proportion, the whole series as- 
suming the form of a general retrogressive curve, 
having hyperbolic angles like those of the Satur- 
nian hyperbola. But, by increasing the number 
of angles allowed to a branch, the lengths of the 
successive branches may be drawn out according 
to various rules, or even with perfect freedom, 
while their regular deduction as to quality is pre- 
served. 

This new mode of composition was probaby the 
one which belonged to the character of Pluto, 
the ruler of the shades, to whom also may be 
ascribed the power of exploring internal posses- 
sions. To this mode of composition may also be 
referred the power of search into antiquities, be- 
cause in exploring a retrogressive series it re- 
mounts to those earlier and slower forms which 
must have been anterior, according to that rule. 
Thus Pluto, contemplating his possessions in the 
infernal regions, was enabled to remount to the 



PLUTO OR ADONIS. 83 

earliest generations which had been upon the 
earth. This power is also capable of a sort of al- 
ternation by each branch crossing itself, and 
changing its direction towards the inner side of the 
general series, on which principle the whole series is 
enabled to preserve its general course as an invo- 
lute. The first branch, by crossing, will place the 
concave side outwards ; the next branch, by cross- 
ing, will again turn out the convex side ; and so 
on alternately. Thus Acheron, the river of pain, 
so named perhaps from drawing out particular por- 
tions of form, continued its descent through the 
infernal regions. This mode of composition has 
most relation to the elementary power of Mars, 
because of its perfect freedom of changing its 
course in every direction ; not only because of the 
general curve being an involute having an infinite 
series of branches, each of which has a direction of 
its own, but also because of the power of draw- 
ing out the length of any particular branch, by 
which the position of the branch that is to follow 
may be totally altered, and by which there is a 
perfect freedom of giving it any direction. This 
is like the power of communicating ductility to 
forms ; and, in the celebrated hymn in honour of 
Proserpine, which is said to have been sung by 
Pindar appearing after his death, there was ap- 
plied to Pluto the ephithet of K^vrnms, the gold- 
en. To this class may be referred some of the 



84 PLUTO OR ADONIS. 

greatest painters who have excelled in gracefulness, 
like Raphael; because the source of freedom of 
design is the power of shortening some parts of 
forms while others are increased or drawn out. To 
the same class may be referred Rubens, who did 
not excel in delineating beauty, but who had the 
same perfect freedom of changing the course of his 
lines, and making the figures appear as if turning. 
Another form of this power was Adonis, the be- 
loved of Venus. 

Among the ancient nations, the character of 
the Arcadians seems to have been that which came 
nearest to this mode of composition. Living in an 
inland country, they passed their time in a state of 
rural freedom, enjoying the pleasure of contem- 
plating their internal possessions of flocks and 
herds, and easily changing their intentions as they 
wandered from mountain to valley. A troop of 
Arcadians was present at the siege of Troy ; and 
they are mentioned by Homer as skilful in war. 
But, like Pluto, they preferred nothing to their 
own region, and those inland solitudes, which were 
called by the poets nigri colles Arcadia?, " the 
dark hills of Arcadia." Pan also was said to fre- 
quent them ; but this was not because the charac- 
ter of the Arcadians coincided with his own power, 
but only because of a certain similitude between 
them ; as the power of Pan, which is straight lines 
diverging from a centre, has an obvious relation 



PLUTO OR ADONIS. 85 

to change of direction, and so far coincides Math 
the power of Mars. The regions frequented by 
the ancient divinities were generally not peculiar- 
ly their own, but merely had some partial coinci- 
dence with the power which belonged to them, and 
therefore, were supposed to be agreeable places of 
resort for them. Thus Bacchus frequented Rho- 
dope, and the cold mountains of Thrace belonging 
to Mars. The graces dwelt at Orchomenus in 
Bceotia; and Sicily was called sacred to Proser- 
pina, although it had more relation, perhaps, to 
the power of Ceres. To Pluto may be ascribed 
also the power of acceleration ; because the effect 
of the addition made to the later line in the series 
of proportionals, is to carry out the curve to a 
greater extent or circuit than the same number of 
angles would have produced if the former propor- 
tion of the lines had been continued. For al- 
though each successive branch, to which the same 
number of angles is allotted, is shorter than that 
which preceded it, nevertheless each new branch, 
by altering the course of the curve into a wider or 
more circuitous form, has a real power of making 
the extent of the curve have an increasing propor- 
tion to the same number of the angles which trace 
it. Thus Acheron may be conceived as rolling 
its floods with increasing vehemence, but, at the 
time, with perfect freedom and fluidity of parts. 
The characteristic of this power is internal profu- 



86 PLULO OR ADONIS. 

sion and affluence. Thus Rubens drew a great 
picture representing the fall of the damned. To 
the same class with Pluto may perhaps also be re- 
ferred Pythagoras, one of the greatest among the 
ancient philosophers, and particularly curious about 
the powers of finite numbers. This power, from 
being like a retrogressive form of the Saturnian 
curve, corresponds also with the power of gloomy 
imagination searching into the depths, and was 
shewn in the poetry of Milton, who chose Satan 
for his hero ; for Mars must always be rebellious, 
and wish to have the free prospect of the whole 
universe to himself. This seems also to have been, 
the character of the ancient Persians, who, like 
Milton, delighted in the conception of an evil prin- 
ciple opposed to the good. 

Supposing the additions made to the later line 
in the series were too great, or according to too 
slow a series of proportionals, the length of the 
later line might ultimately become equal to that of 
the preceding one ; and, in this case, the curve, be- 
ing no longer conducted by lines of unequal pro- 
portion, would cease to bend as a retrogressive hy- 
perbola, and would become part of a circle. This 
was perhaps signified in Styx, which was consider- 
ed as only an aestuary of Acheron. Styx was 
sworn by, because the angles, if continued, will 
complete and fulfil the circle, and, unless by a 
farther addition, there is no escape from it. This 



PLUTO OR ADONIS. 87 

name was also given to a lake in Arcadia. If the 
lengths of the two equal lines were continually di- 
minished by the same quantity being taken from 
both, it would give a type of the wearisome and 
stationary power of Orcus, and the series of angles 
would continue tracing portions of circles always 
smaller, and descending to infinitude. Pirithous, 
who attempted to carry away Proserpina, and who 
afterwards was bound in the infernal regions, re- 
presented this power. When he asked what was it 
o^clock, the answer was, eternity. This was also 
the power of Circe, who had the power of chang- 
ing the human species into brutes, which signifies 
making their nature stationary and unprogressive* 



88 



SECT. III. 

ON THE EVOLUTION OF DIMINISHING HYPERBOLIC 
BRANCHES. 

Perseus. 

If, in the series of angles which trace the curve, 
the later line had at last become equal to the pre* 
ceding one, then, if length still continued to be 
added to the later line, it would become the long- 
er of the two, and the series of angles would be- 
gin to trace an evolving curve, which would also 
be the hyperbola. And, if always, after a certain 
number of angles, another addition were made to 
the later line, but, according to the same series of 
quantities, diminishing in continued proportion as 
formerly ; in this case there would be produced a 
succession of evolving branches, for ever expanding 
and increasing in length. This power is derived 
immediately from that of Pluto, and is produced 
by the continuation of the same series of added 
quantities diminishing in continued proportion. It 
was signified in Demogorgon, of which the Arca- 
dians were afraid to pronounce even the name. 



PERSEUS. 



89 



The chief representation of it, however, was in the 
characters of the furies Megaera, Alecto, and Ti- 
siphone, the daughters of Acheron and night ; and 
also, perhaps, it belonged to Cocytus, the river 
of lamentation, because of its evolving form ; for 
it was understood that Acheron flowed into Co- 
cytus. Thus the power of Pluto may come to an 
end and be lost in that of Cocytus ; from whence 
the ceremony of weeping for Adonis. 

But as this series would have no limit to the 
increase of its size, it becomes necessary to sup- 
press and restrain the expanding power, by trans- 
posing the angles which trace each successive branch 
into a smaller scale, for instance a fourth. This 
will make the branches continually diminish their 
length, with immense rapidity, though they con- 
tinue evolving, and pursuing the same deduction 
as before. This mode of composition was proba- 
bly signified in the character of Perseus, who slew 
Medusa, and placed on his shield the gorgon's 
visage, surrounded with snaky hair, and still re- 
tailing sufficient terrific powers to convert the 
spectators into stone. But, this mode of compo- 
sition, by the successive changes of scale, passes 
into infinite refinement, and acquires another pe- 
culiarity, which is, that the branches, by their suc- 
cessive diminutions, turn in, and change the course 
of the whole series so fast that it becomes an invo- 
lute, while all the particular branches continue 



90 PERSEUS. 

evolving. This is like those powers of magic by 
which rivers were made to flow back towards their 
sources. The characteristic of this mode of com- 
position is, that it is like a vast extent gathered up 
within narrow limit s, because each branch in the 
diminishing series represents one which would have 
been found at an immense distance, and situated in 
a totally different curve, if the original series had 
been followed out, without transposition into a 
smaller scale. But that order is reversed ; and in 
proportion as the branches would have been far 
off and of great extent, they are represented by 
branches more minute and delicate, and farther 
withdrawn into the bosom of the curve. The cha- 
racter of the Spartans was that which came nearest 
to this mode of composition, because they were 
distinguished for brevity of expression, for theft 
and concealment, and for great courage and power, 
which seemed ridiculously to defy what they were 
totally disproportioned to. Perseus was called the 
grandson of an Argive prince, but had no relation 
to the Argive region, and may be assumed as the 
type of the Lacedaemonian character. This mode 
of composition has most relation to the elementary 
power of Mercury ; because it partakes of the nature 
of a series of evolving hyperbolic branches, such 
as exists in the Saturnian curve. One of the attri- 
butes of Mercury was speed in going to distant 
places, and the winged sandals were also ascribed 



PERSEUS. 91 

to Perseus. Mercury was also the protecting god 
of thieves. To this mode of composition may also 
be referred the character of Somnus, the brother 
of death, with the power of imagination; since 
the curve of Perseus resembles the Saturnian hy- 
perbola, to which imagination originally belongs. 
The muses, in searching out the different forms of 
the Saturnian hyperbola, virtually explore the dif- 
ferent modes of imagining, and contrast them ; but 
the power of Perseus belongs all to one curve, and 
is like the infinitude of one vast and endless dream. 
This character may be ascribed to the ancient 
Danes, among whom the mythological poems of 
the north were composed. This mode of compo- 
sition may also be made to alternate, by each 
branch crossing itself and changing its course to- 
wards the inner side of the general series, by which 
means the whole together will preserve its course 
as an involute. The first branch, by crossing, 
will place the concave side outwards; the next 
branch, by crossing, will again turn out the con- 
vex side, and so on alternately. This becomes the 
same as the consciousness of sleep, or a sublime 
feeling of mental illusion, which some of the eastern 
sages endeavoured to cultivate. Mercury was called 
the son of Maia ; and he may be conceived sliding 
through the region of the air, as if lost in dream. 
To this mode of composition belongs a bold and 
oriental mode of imagining, and a style of expres- 



92 PERSEUS. 

sion passing into hyperbola and metaphor; since 
metaphor is the substitution of another form for 
that which was originally meant. This power is 
not so much allied to the feelings of modern 
Europeans, such as they have hitherto existed. 
Among the English poets, Coleridge seems to 
come nearest to it In another point of view, this 
mode of composition is the power of bringing near 
that which is remote, as if the vision of Perseus 
extended to the summit of Olympus, or to the re- 
motest of the glittering tents and pavilions of genii 
spread in the air, and not appearing till after sun- 
set. The true sage sees things afar off; he seems 
as if he could lean the arm of his understanding 
on the horns of the moon ; he has nine hundred 
and ninety-nine ears for every sort of instruction ; 
the thousandth is reserved for sleep, and, in the 
daytime, for the habitual songs of the grasshopper, 
or the murmur of a neighbouring brook. The want 
of self-constraint allows troubles to expand, and 
leads to a state like the whirl of dancing santons ; 
but surely the virtuous man enjoys peace ; and the 
only things which he covets are those which belong 
to time and distance. In the fine arts, that which 
seems most connected with this mode of feeling is 
relievo, in which the dimensions of solid figures 
are made to approach to the nature of a plane ; 
because this is like the power of contraction and 
compression which belongs to the Lacedaemonian 



PERSEUS. 93 

character. If there were Spartan artists in ancient 
times, it is probable that they would have a pecu- 
liar talent for works in relievo. The nature of re- 
lievo, however, would be coarse, and would have 
nothing sufficiently peculiar to itself if the figures 
were too prominent ; and its greatest delicacy is 
when the figures seem almost to sink and vanish 
in the plane; for, since its origin is from solid 
figures, it is evident that the boldest relievo must 
be, in reality, that which makes them approach 
nearest to a plane, where they appear as if about 
to vanish. This mode of composition was also the 
peculiarity of Horace, in whose odes is shewn the 
power of contraction bringing together materials 
from great distances animoque rotundum percur- 
risse polum, geographical references, which often 
compress a vast extent of dominions within the 
compass of a single ode. Jerusalem was built in 
the region which belonged to the tribe of Benja- 
min ; and, perhaps, the Jewish style of architec- 
ture may have had some relation to this mode of 
composition, in which the proportions are not of a 
direct and simple meaning, but in proportion to 
their decrease expressive of something farther off, 
and much greater in size. Such was Nemesis, 
who reversed the fortunes of the great and proud. 
This was perhaps also the character of the ancient 
Chaldeans, who delighted in observing the distant 
phenomena of the heavens. 



94 



SECT, IV. 

ON THE RETROGRESSION OF THE HYPERBOLA 
THROUGH A SERIES OF INCONSECUTIVE FORMS. 

Vertumnus or the Fates. 

But, differing from all these powers, and existing 
entirely apart from them, a new retrogressive series 
will be found, if in the succession of angles as they 
existed at first in the power of Proserpina, the 
later line instead of being added to, as in the power 
of Pluto, at each new branch, be, on the contrary, 
diminished at the beginning of each new branch, 
according to a series of quantities diminishing in 
continued proportion, so as never to exhaust the 
extent of the later line. In this case, the general 
series, instead of passing always into slower forms 
of the hyperbola, will make its transitions into 
more rapid forms. These exist together in an 
order which is the reverse of that which belongs 
to the original form of the Saturnian hyperbola, to 
which the curve of Pluto has a certain resem- 
blance ; because the curve of Pluto coincides with 
the natural order or derivation of branches, each 



VERTUMNUS OR THE FATES. 95 

leading to a slower form which had preceded it. 
But, when the later line, instead of being added 
to, is diminished, then the points at which the 
branches are joined together must all project to- 
wards the convexity of the curves ; and each 
branch becomes like a terminated form, not natu- 
rally leading either into that which precedes, or that 
which follows, so that it is situated as if it did not 
belong to any general curve, but was only contiguous 
to those forms beside it. This mode of composition 
appears to have been signified in the characters of 
the Fates, the daughters of Erebus and night. 
Their business was to determine the limits of time, 
fortune, or any thing else, to individuals ; and this 
corresponds with the power of searching out par- 
ticular modes of extension, defined and separated, 
as it were, from those which precede and follow 
them. To Clotho, Lachesis, and Atropos, may be 
ascribed also the power of retardation ; because, 
in each new branch, the same number of angles 
will trace a form of smaller extent or circuit than 
would have been found if the same proportion of 
the lines had been continued. Therefore, in the 
succession of branches, the extent or circuit of the 
curve, has a decreasing proportion of the same 
number of angles; and this is the same as the 
power of retardation. This power must be sup- 
posed to alternate in a manner peculiar to itself; 
each branch crossing itself once, and terminating 






96 VERTUMNUS OR THE FATES. 

in a form which is crossed by the succeeding branch 
springing from it. Thus Lethe, the river of obli- 
vion, passing through the deepest shades, soothed 
them with the changing current of its waters, which 
appearing to stop in successive eddies, always flow- 
ed on in a new direction. This mode of compo- 
sition has most relation to the elementary power of 
Vulcan or definition, because of its relation to ter- 
minated forms. Of the character of the ancient 
Argives nothing is now known but that they were 
considered as the favourite people of Juno. To 
the Fates may be ascribed the sense of contrariety 
in those things which are united at a common 
boundary. 

One of the greatest poets of antiquity is said to 
have been Palamedes the Argive, who was co- 
temporaneous with the Trojan war, and from whom 
Homer is supposed to have borrowed many things. 
But to the same class with the Fates may be re- 
ferred Pindar, whose poetry, besides celebrating 
the good fortune of individuals, relates mostly to 
the peculiarities of different countries, divinities, or 
races of mankind, and corresponds with the power 
of defining, or of specifying, what is peculiar to 
each, and what each obtains. The Fates had the 
task of ascertaining what lots or conditions were to 
fall to individuals, and Vulcan may be conceived 
as intellectually discriminating the natures of ab- 
stract powers. To the Fates may also be symbo- 



VERTUMNUS OR THE FATES. 97 

lically ascribed the parrot as a type of articulation ; 
because each vocal power, in the structure of words, 
is like an abstract rule which may be prolonged to 
any extent, before the vowel is closed or shut in ; 
but the knitting of these different vocal powers to- 
gether is articulation, and corresponds with Vul- 
can, or the conjunction of different abstract rules in 
that which is one ; for it is essential to articulation 
that all the component vocal powers be limited. 
Among the Asiatic nations, the mode of composi- 
tion belonging to the Fates seems to have been 
exemplified in the Phoenicians, who, like the Ar- 
gives, were a mercantile people, dwelling upon the 
sea coast, and skilled in maritime affairs. Thales 
is said to have been born at Miletus, but his pro- 
genitors were Phoenicians, and he was understood 
to belong to that nation. In modern times, the 
same character seems to have belonged to the Ve- 
netians, also a mercantile people, dwelling on the 
border of the sea ; and the peculiar talent shewn 
by their painters, who excelled in colouring, also 
corresponds with the power of Vulcan, because the 
play of each simple colour is one abstract rule, and 
the art of blending colours is the same as the con- 
junction of different rules in that which is one, or 
the conjunction of different straight lines in one 
triangle or polygon. To the Fates may also be 
ascribed the power of observing the change of 
times, and of marking the boundaries of periods 



98 VERTUMNUS OR THE FATES. 

differing in kind ; and this was perhaps signified 
in the character of Vertumnus, the god of the sea- 
sons. To this class may also be referred the osten- 
tatious peacock, the favourite bird of Juno, assign- 
ed to her by hypothesis or imputation. The Ve- 
netians were characterized by a taste for buffoonery ; 
and this also belongs to the Fates, whose portion 
is to know the definitions and properties of all the 
passions, and their respective tendencies ; and this 
is the source of pantomime, which, in a philoso- 
phical point of view, is the source of the most re- 
fined intellectual pleasure. But mankind are gene- 
rally too confined in their views, and too much un- 
der the influence of particular affections, to perceive 
the advantage of any such general taste, and they 
most frequently are inclined to regard buffoonery 
as an insult to the particular feelings which rule 
in their minds for the time, or which habitually 
belong to their characters as individuals. This, 
however, shews the excellence of the powers of de- 
finition as a source of reproof to the ignorant and 
selfish, and as a commemoration of the magnificent 
variety of powers which exist in the universe, and 
which, in ancient times, was exemplified in the as- 
semblage of the different nations at the Olympic 
games. 

Of those four powers which are produced from 
the application of the power of Diana, or angle, to 
the hyperbola, the Fates, being disjoined and apart 



VERTUMNUS OR THE FATES. 99 

from the rest, should be placed first, then the mode 
of composition which belongs to Proserpina, then 
Pluto, and last the power of Perseus, in which the 
particular branches all resume the evolving form 
originally belonging to the hyperbola. 



100 



CHAP. V. 

ON THE INTERMINABLE FORMS OF TRANSVERSE 
PROGRESSION. 



SECT. I. 

ON THE POWERS OF CONTINUITY AND SYNTAX. 

Erichthonius or Hyperion. 

In the foregoing modes of composition, a series of 
angles having been already applied, the applica- 
tion of the next power, Vulcan, to the curves, 
which are guided by angles, will only have the ef- 
fect of making the branches cross, and produce a 
new form of the transverse mode of progression, 
which belongs to Bacchus. The power of Vulcan 
may be applied to such a form as that of Perseus * 
and each branch may cross by one of the straight 
lines in the series of guiding angles being caused 
to turn back, and make the complement of the same 
angle on the other side of the preceding straight 
line. This change of course in the middle of 



ERICHTHONIUS OR HYPERION. 101 

each branch would make the whole series quit the 
form of the general curve of Perseus, and stretch 
forward by alternating, as in the mode of progres- 
sion which belongs to Bacchus. But, in this case, 
there being no addition to the thickness of the suc- 
cessive branches, the series would never be stopped, 
but would extend to infinitude, and is the same 
as interminable continuity. 

This mode of composition has most relation to 
the elementary power of Apollo ; and it was per- 
haps signified in the character of Hyperion, the ce- 
lestial charioteer, and also in Erichthonius, the in- 
ventor of the chariot ; which is the same as draw- 
ing or the continuity of a line, in which the ad- 
vancing point retains its connexion with the ex- 
tension thrown behind. The mode of composition 
nearest to this is a branch returning across itself 
in the direction contrary to its own previous ex- 
tension, and so blending contrary tendencies into 
one form, which appears in the intersections of the 
parallel lines, as in the mode of composition which 
belongs to Bacchus ; but the power of Bacchus 
consists always of finite series of branches ; while 
this mode of composition which belongs to Erich- 
thonius consists of one endless series, and therefore 
corresponds with the infinitude of continuity. In 
a straight line, any two points are contraries, but 
they are blended into one by a perfect intermediate 
continuity of extension, in the same manner as 



102 ERICHTHONIUS OR HYPERION. 

the horses are bound to a chariot by the traces ; 
and thus Erichthonius continues for ever uniting 
contraries in the form of each successive branch 
which crosses. 

In this mode of composition, each successive 
branch, as in the form of Perseus, may be trans- 
posed into a smaller scale, to prevent the excessive 
expansion of the form, which would otherwise take 
place as formerly in the Gorgons ; and which was 
probably signified in the giant Typhon, a furious 
whirlwind, whose powers at first terrified and dis- 
concerted all the gods. Or thus the inconsiderate 
Phaethon, having obtained from Phoebus the boon 
of being permitted to drive his chariot, could not 
sufficiently restrain or control the horses, and was 
struck from his car, to prevent the conflagration 
of the heavens. But by the successive diminutions 
of scale, this form contracts and p asse into infinite 
refinement, so that the added branches, becoming 
always less, scarcely make any visible addition to the 
extent of the whole form. This mode of composition 
being the same as the elementary power of Apollo, 
partakes of the nature of a list or catalogue of par- 
ticulars, and becomes the same as genealogy and 
the power of determining the order of particulars 
as to priority in time or place. To Erichthonius 
therefore may be ascribed historical narration; 
and this character seems to have belonged, among 
the ancient nations, to the Egyptians, who were 



ERICHTHONIUS OR HYPERION. 103 

remarkable for retaining among them the memory 
of the past, and for continuing the same manners 
and opinions which they had derived from the re- 
motest antiquity. The Egyptians also excelled in 
construction, which is the art of binding together 
different particulars in the same continuity ; and, 
therefore, construction, and the power of following 
out a method which has been begun, are always 
found together ; and to this class may be referred 
those mathematicians who study the generation of 
any simple curve as a list or genealogy of points 
bound together in one continuity, and having their 
places according to a certain rule. Every abstract 
rule changes and passes through successive states 
as it is continued, and, therefore, Erichthonius is 
also the power of metamorphosis^ or the interme- 
diate connexion between a state or condition which 
has been left and another which has been passed 
into. Thus Ovid, at the beginning of his poem, 
asks the gods to enable him to deduce a perpetual 
song from the beginning of the creation. In the 
Scripture, the power of genealogy and historical 
narration is shewn in Moses, who belonged to the 
tribe of Levi, and who was lifted from the waters 
of the Nile. To Erichthonius must also belong 
the syntax or construction of words in language. 
To the same class with the Egyptians may per- 
haps be referred the European Thebans, deriving 



104 ERICHTHONIUS OR HYPERION. 

their origin from Cadmus and Hermione, who were 
changed into dragons^ that is, powers of continuity. 
But both Bacchus and Hercules were born in 
Thebes, from its relation to the transverse mode 
of progression. Pindar and Hesiod, although 
born in Bceotia, were both foreign to the Theban 
character, which rather belonged to Herodotus. 
The proverbial dulness of the Boeotians may have 
arisen from too much tenacious continuity. To 
this class may be referred the duck, the goose, and 
the swan, which are similar in their character to 
the protracted dragons of Bceotia ; and the swan 
was sacred to Apollo. But the same power was 
perhaps also exemplified in the Rhodians, who 
seem to have worshipped Apollo as a sort of ever- 
lasting Bacchus. Aristophanes, a native of Rhodes, 
was an instance of that power of ridicule which de- 
pends upon the art of forced construction, or the 
union of contrary tendencies. Thus the horses of 
Erichthonius, in first getting into motion, seem as 
if they were going to leave the chariot behind 
them, but it is compelled to follow. To this power 
may also be ascribed sophistry and paradox, as ex- 
emplified in Hume, w r ho was also an historian, and 
Rousseau, one of whose favourite studies was edu- 
cation, in which the mind is carried on through 
successive states or conditions ; and perhaps the 
character of the Swiss, in general, may be founded 



ERICHTHONIUS OR HYPERION. 105 

on the same power of sustained continuity which 
belonged to the ancient Egyptians. Regularity of 
habits made the Swiss nation much esteemed as 
servants throughout Europe. This is also the 
character of Ganymede, the cupbearer of Jove ; but 
Hebe was a form of the limited series which be- 
longs to Bacchus ; and, in reference to its termi- 
nation, she is said, in bearing the goblet, to have 
stumbled before the assembled divinities, and dis- 
pleased them ; after which Ganymede was made 
cupbearer in her stead. In Asia, the same char- 
acter seems to belong to the Chinese, who are 
equally tenacious of habit, while the progress of 
time carries it always into greater refinement. The 
Theban Niobe, boasting that she equalled Latona 
in beauty, had her children shot by Apollo and 
Diana; and, in long weeping and refining upon 
her grief, was changed into a stone. To this 
power may also be ascribed the power of refining 
the quality of one substance or material, as exem- 
plified in the Chinese pottery, and the colours 
given to it ; for these shew the art of refining 
single hues, which are as single rules or powers of 
continuity. And Titian, in painting, may be 
classed rather as a Swiss than as a Venetian, since 
it is the opinion of the skilful, that his chief ex- 
cellence is in the clearness and strength of single 
hues, such as that of the flesh. Among the 

e 2 



106 ERICHTHONIUS OR HYPERION. 

apostles, the power of Erich thonius seems to have 
belonged to St Matthew, who is the chief of the 
Evangelists as to history ; and who was of the 
tribe of Levi, and called from sitting at the receipt 
of the custom. But the chief characteristic of this 
mode of composition is self-will and reliance on the 
power of habit ; from whence its relation to edu- 
cation. 



107 



SECT. II. 

ON THE DEDUCTION OF COMPOSITE HYPERBOLIC 
BRANCHES. 

Theseus. 

The application of the power of Vulcan also pro- 
duces another kind of infinite series, which is part- 
ly similar to that of Erichthonius, but essentially- 
different in power. For if the crossing, instead 
of taking place in the middle of each branch, be 
employed always at the point where the last 
branch terminates and the new one begins, in this 
case the transverse form of progression will still 
be continued ; but each hyperbolic form will in- 
tersect the parallel courses of the last branch. 
Therefore, in this deduction, the parallel courses 
of each branch, instead of being crossed, as former- 
ly, by a farther part of the same curve, will be 
measured against a new form, entirely separate 
and distinct. This mode of composition is the 
same as the power of transition into forms belong- 
ing to a different rule or generation ; because the 
forms which cross are not like those in the power 



108 THESEUS. 

of Bacchus or Erichthonius, merely different parts 
of the same curve or rule ; but, on the contrary, 
they exemplify the power of arbitrary composition, 
in bringing together forms which are not continua- 
tions of the same rule. This mode of composi- 
tion, therefore, has most relation to the elementary 
power of Minerva, or arbitrary composition ; and 
it was probably signified in Theseus, the hero of 
Athens. The characteristic of this mode of com- 
position is the same as intellectual freedom, and 
agility in escaping from deduction, or from the 
remainder of what belongs to the same rule. 
Therefore it becomes as wit, which disconcerts 
the powers of tediousness, and will not long bear 
with sameness of method. But, in regard to the 
blending of powers foreign to each other, one 
characteristic of the Athenians was their love of 
strangers, and the encouragement which foreigners 
had to come among them. The number of famous 
characters really born in Athens was small in 
comparison with the number of those who only 
resided there. Theseus himself was born in the 
Peloponnesus. Solon was a native of Salamis ; 
Diogenes was a native of Sinope ; Xeno came 
from Cyprus ; Aristophanes from Rhodes ; Aris- 
totle was a Macedonian ; Theophrastus a Lesbian; 
Chrysippus was born in Cilicia ; and to this list 
might be added many more. These persons were 
cherished by the Athenians, as if for their own 



THESEUS. 109 

taste in having among them forms entirely remote* 
and incapable of having been derived from the 
same source. Among the modern nations of 
Europe, the French approach most to the charac- 
teristics of this mode of composition, and shew 
the same talent for considering the temporary 
union of forms with others, not having a common 
derivation. This is the power of analysis, as 
shewn in Samuel Johnson, Jeffrey, and other 
modern critics. To Theseus may also be ascribed 
the love of transitions, which, by this mode of 
composition, are found without number, in a de- 
duction which extends to infinitude. Thus, by 
a stroke of the hammer of Vulcan, there was 
brought forth the immortal or interminable 
Athana, clad in armour, and also bearing her 
shield as a superabduction. The name Pallas, 
signifying vibrating, may apply to the blending 
of two different motions or rules. To this mode 
of composition may also be referred doubt or in- 
credulity, because of the difference of the forms 
which are blended. These do not belong to the 
same rule, or spring from the same source ; and, 
therefore, their union is the same as the feeling of 
ambiguity, as shewn in the character of St 
Thomas, whose name signifies Geminus, double. 
Scepticism, according to some, was the distin- 
guishing characteristic of the atomical Democritus. 
A sense of the presence of different elements in 



110 THESEUS. 

that which appears as one leads to analysis, or the 
inclination to separate and discriminate them ; 
from whence the intellectual character of Pallas, 
whose wisdom was the same as the power of re- 
solution, and corresponds with the science of che- 
mistry. Her favourite bird the owl assumed its 
position in some ivied niche or dilapidated win- 
dow of an uninhabited tower, and, during the 
silence of night, seemed tranquilly to enjoy the 
pleasures of speculation or scepticism. Its eyes 
were directed to the ground, where, by the moon- 
light, it distinctly perceived every blade of grass 
that entered into the composition of the matted 
sod, or received a foreign lustre from being gar- 
nished with a drop of dew. Perhaps the fox cau- 
tiously glided past, as if trusting that the sound 
of his steps could not be distinguished or separ- 
ated from the murmur of a distant torrent which, 
being lost among chasms of rocks, seemed unwill- 
ing to announce that it retained any activity dur- 
ing the hours of repose, and rather moderated and 
restrained its voice, as if in soothing complaisance 
to the rest of the world. But, in the meantime, 
the favourite of Pallas remained awake and atten- 
tive to the sounds that were stirring, never failing 
to solve and discriminate those which occurred to- 
gether. The last result of analysis is the sense 
of vacancy, negation, or silence, after all the ele- 
ments in a compound have been exhausted or 



THESEUS. Ill 

taken away. In reference to this, the Athenians 
stamped upon their coin the figure of an empty 
jar, lying upon its side, to shew that the water 
which it had contained was all withdrawn. The 
reason is, that a composition has no element which 
is peculiar to itself, or which remains and consti- 
tutes its essence ; and, therefore, when the ingre- 
dients are separated, the result is negation, or the 
composition ceases to exist. This sublime feeling 
of silence and negation belongs peculiarly to Mi- 
nerva or Theseus, and is connected with scepti- 
cism, to which there belongs a refined intellectual 
pleasure. The application of synthetical power to 
the fine arts produces a firm style, in which the 
component elements are not concealed or disguised, 
but allowed to appear, as in antique statues, of 
which the lineaments are not soft and natural, but 
rather composed of parts exemplifying different 
rules, like separate mathematical forms. Hence 
the style which belongs to the power of Minerva 
is founded on the beauty of such composition as is 
visible and intelligible. This was probably the 
character of Apelles, whose style in painting 
pleased the ancients most. The same is the case 
in architecture, in which the forms united are 
foreign to each other, and evidently generated 
from different rules. To the same class with 
Theseus may be referred the Oreades, or moun- 
tain-nymphs ; because every hill is a form acci- 



112 THESEUS. 

dental or extraneous to the general level of the 
region in which it is situated ; nor is the mountain 
"a continuation of that form which belongs to the 
neighbouring plains. It may be doubted whether 
the true Athenian character belonged to Plato ; 
for his compositions, instead of exemplifying ra- 
pidity, fire, and freedom of transition, are for the 
most part excessively protracted and tedious, and 
seem rather to belong to the power of Erich- 
thonius, or continuity, which is more allied to long 
arguments. Socrates was accustomed to fasten on 
some proposition acknowledged by the person with 
whom he was conversing ; but Socrates seems to 
have been a form of Pan or repetition. Nor 
should this excite astonishment, for it is probable 
that even the universal Shakspeare, distinguished 
for his knowledge of the more vulgar constitution 
of human nature as fixed and unprogressive, was 
no other than a form of the universal Pan; to 
whom belongs rotatory motion, and the repetition 
of cycles. 

From continuity like this the power of Theseus 
must always seek to get free ; because the charac- 
teristic of Pallas is the arbitrary union of forms 
derived from totally different sources, and only 
allied by accidental situation, in which they still 
remain foreign to each other. This talent was 
shewn in Voltaire, whose agility was like that of 
Myrinna, whose tomb was an hill celebrated by 



THESEUS. 113 

Homer as the place where the Trojans drew out 
their troops. The characteristic of Voltaire was 
freedom of transition. Thus, he describes a per- 
son travelling from world to world through the 
aerial medium, and struck with astonishment at 
the dissimilarity of the modes of existence which 
he is thus enabled to compare and measure against 
each other. He receives from a learned man a 
book which is to teach him the last results of all 
inquiry, and when opened it is found to be an 
universal blank ; which is like the negation found 
by analysis, or by successively taking away the 
constituent parts of any composition. The power 
of analysis also gives birth to clearness and per- 
spicuity of arrangement. This characteristic was 
shewn in music by Haydn, whose mode of com- 
bining is such as to exhibit the elements of musical 
form in the utmost clearness, but in an order 
which does not sooth or flatter the ear by much 
continuity. The music of the ancient Athenians 
must have had most resemblance to that of Haydn, 
which is the result of analysis, and presents the 
elements of musical form, as it were, perspicuous- 
ly and apart. To the power of Minerva or The- 
seus there may also be ascribed, as a type of agili- 
ty, the monkey, which, in its natural state, lives 
among the branches of trees, and leaps from one 
to another so easily, that it is capable of passing 
through a whole forest without ever touching the 



114 THESEUS. 

ground. The reverse of this was shewn in An- 
taeus, who gathered new strength every time he 
touched the earth, and who, like David Hume, 
seems to have been a form of Erichthonius or con- 
tinuitv. 

In this mode of composition, as in the preced- 
ing, the angles which guide each successive branch 
must be supposed to be produced on a smaller 
scale to prevent the expansion of the form, which 
would otherwise become unlimited. The fall or 
contraction of breadth must be supposed to take 
place in the middle of each branch, and the same 
breadth to be continued to the middle of the next 
branch ; so that there will be no change of breadth 
at the intersection of the two branches. The un- 
limited expansion of this form was perhaps signi- 
fied in the character of Orestes, who, having put 
to death his mother, was thenceforth deprived of 
repose. His situation became a favourite subject 
for tragedy, and the unhappy criminal, flying 
across the stage, was seen pursued by the furies 
with their scourges of snakes. When continued 
extension passes into another rule, it in a manner 
destroys or terminates the rule or mode of exten- 
sion which precedes it ; and the ancients seem to 
have thought that such a transition resembles 
parricide, because, in theory, it makes an end of 
the previous generation ; but Solon made no law 
against parricide, he said because he disbelieved 



THESEUS. 



115 



in that crime. But, by the transitions of the 
branches into smaller scales, this form passes into 
infinite refinement, and the successive branches, 
which increase its extension, contract and diminish 
so fast, that the whole deduction scarcely appears 
to advance ; while, at the same time, it continues 
to trace out an interminable series of composite 
forms ; and Pallas, like Perseus, wears the head 
of the Gorgon upon her shield. Theseus is the 
same as the Ephraim of the Jews, and Hercules is 
Judah. 

To this mode of composition must be referred 
the philosophy of Locke and others, as to the as- 
sociation of ideas ; which is similar to the ancient 
Epicurean philosophy, both as to feeling and as to 
theory ; in the first place, as to feeling, because 
the sense of enjoyment which it cultivates relates 
to those things which are agreeably assembled for 
the present ; and it relates neither to the past nor 
the future, since it is not the prolongation of any 
one rule or continuous form, but is like the sense 
of ingredients well blended, and producing the 
state or condition of the mind for the time being. 
In theory, the association of ideas, according to 
Locke, and the concourse of atoms of different 
kinds and figures, according to the method of Epi- 
curus, evidently belong both to composition. The 
discriminating and separating of these according 
to their supposed natures is a sort of analysis. 



116 THESEUS. 

But the power of analysis leads always to that 
sense of vacuity or negation which Locke sup^ 
posed to exist in the mind previous to external 
impressions. But Theseus, after having found a 
silence, or vacuity, again resorts to composition, 
and reassembles the scattered elements. To The- 
seus may also be ascribed the assemblage of mu- 
tual auxiliaries, a characteristic which appeared in 
all the achievements of him as the power of syn- 
thesis. It is perhaps also the Chimaera, an union 
of forms without real continuity, but breathing 
fire, because the transverse mode of progression 
has a certain relation to the power of Vulcan. 
This power must always want the freshness of 
nature and life ; but, at the same time, it has ad- 
vantages which distinguish it from all the other 
powers. Therefore Horace, after Jove, ranks 
Pallas next, Proocimos illi tamen occupavit Pallas 
honores ; and, in Greece, the Athenians were con- 
spicuous beyond the other nations. 



117 



CHAP. VL 

ON THE PROGRESSIONS OF DOUBLE FORMS. 
SECT. I. 

ON THE POWERS OF FLUCTUATION. 

Pollux or Eridanus. 

The next elementary power to be applied is that 
of Neptune, which has two angles, or two planes 
extending at the same time. This power will 
have the effect of resolving such a curve as that 
of Perseus into two separate parts. One of the 
curves must be the ruling one, and must be 
supposed to be guided by a certain series of 
angles, such as that of Perseus ; and if, on the 
outside, there be made to project from the point 
of each angle a straight line equal to that which 
is the shorter side of the angle, and if the tops of 
these projecting lines be connected by straight 
lines, there will be produced another and outer 
series of angles, running collateral to those of the 
ruling curve, and capable of tracing successive 
branches opposite to those in the ruling curve, 



118 POLLUX OR ERIDANUS. 

Now, since, in this figure, quadrangles are built 
on each line which guides the ruling curve, and 
these quadrangles are all similar figures, therefore 
the uppermost sides of the successive quadrangles 
are in the same proportion to each other, as the 
successive lines which constitute their bases, and 
which also constitute the series of angles guiding 
the ruling curve, which is the inner. And, from 
the structure of the figure, it also follows that, in 
the outer series, the lines are always joined to- 
gether at the same angle as in the inner series. 
Therefore the outer curve consists of successive 
branches, each of which is, in reality, the same 
form as the inner branch opposite to it, but pro- 
duced on a larger scale, and diverging from the 
inner branch. 

But, in this mode of composition, the ruling 
curve should not be indebted for its form to the 
same power as Perseus, but may take a new form 
peculiar to itself; and this is by at each branch 
shortening the later straight line by a quantity 
less in continued proportion, so as to leave the 
later line always greater than the preceding. In 
this figure the successive branches are e volutes, 
but become always slow r er in quality, while, in 
the form of Perseus, they become always more 
rapid. The mode of composition now described 
corresponds w r ith the elementary power of Juno ; 
because, in the two collateral series, the very 



POLLUX OR ERIDANUS. 119 

same form is always found opposite, and in a dif- 
ferent place. Between the two forms relation 
passes to and fro, and like the waters of a river 
between its banks. This is the mythological 
character of Pollux, the god of pugilism, which 
art depends upon the opposition of forms, part for 
part. If the two series of forms were continued 
always on the same scale, they would continually 
diverge ; but, to prevent the excessive expansion 
of the form, it is necessary that each successive 
pair of branches be transposed into a smaller scale, 
for instance a sixth ; and this will greatly change 
the course of the outer series which is diverging ; 
because, the inner series being taken as a guide 
for the other, it is evident that, after each trans- 
position into a smaller scale, the top of the last 
projecting line (which is on a larger scale) must 
be connected with the top of the next (which is 
on a smaller scale), by a line which makes a great 
and sudden descent towards the inner curve, carry- 
ing down the outer curve along with it. After 
this descent the outer curve will be again pro- 
duced, and will diverge as before But, by means 
of the diminutions of scale, the general course 
of the two series will become that of involutes 
passing into infinite refinement. This mode of 
composition is also the mythological character of 
Eridanus or the Po, the image of which is also 
seen flowing among the stars, because it drowned 



120 POLLUX OR ERIDAXUS- 

the burning rays of Phaethon after he was pre- 
cipitated from the chariot of the sun. 

To this mode of composition may also be re- 
ferred pilgrimage, as the fluctuation or change of 
that which passes into another place, like the waves 
of a river. Among the Homeric heroes, this 
power belonged to Ulysses, who visited every 
different shore, and even passed to Cimmeria to 
have conferences with the shades of the dead. 
The crafty wisdom attributed to Ulysses may 
refer to his shewing only one side of his thoughts 
or intentions. But, in relation to novelty of cir- 
cumstances, or the fluctuation which arises from 
the change of distances, the characteristics of this 
power are likewise exemplified in Bunyan^s alle- 
gory of the pilgrim's progress. Pollux was always 
represented having over his head a star, which 
must have been a planet or wanderer. To this 
class must also be referred the mythological char- 
acter of Atlas, king of Mauritania, who was feign- 
ed to support the heavens on his shoulders, be- 
cause Uranus or Coelus is the same as relation, 
or the power of Juno, which belongs to this mode 
of composition. Atlas signifies unshaken, but he 
was probably named by antiphrasis ; and his 
character must have referred to the powers of 
revolution and change in the heavens, by which 
new conjunctions are brought about, because all 
novelty of external circumstances comes originally 



FOLLUX OR ERIDANUS. 121 

from motion. To Pollux and Atlas may there- 
fore be ascribed agitation and change, and the 
celestial sign of the balance, which shews the 
effect of the same weight or power in a different 
place. This was probably the character of the 
ancient Carthaginians ; from a colony of whom 
the Irish nation is supposed to be partly descend- 
ed ; and to the same class may be referred Hymen, 
the god of marriage, because of his relation to the 
power of Juno ; for his torch shines with an agi- 
tated lustre on the affairs of mankind, and has an 
influence like that of the r>lanets over their for- 

JL 

tunes. In music, this power seems to have be- 
longed to Handel, whose compositions have a 
kind of motion and tumult which cannot be imi- 
tated by method ; and, in the fine arts, this power 
is characterized by the expression of fluctuation, 
or by that kind of felicity which appears to result 
from chance. In modern times, it has been shewn 
in the poetry of Sir Walter Scott, which often re- 
lates to peregrination and encounter, or to the 
change of circumstances which is produced by the 
powers of motion going on in the world ; or some- 
times even to the influences of the planets. The 
art of expressing fluctuation in the structure of 
verse belonged, among the ancients, to Catullus, 
the author of the Epithalamium of Peleus and 
Thetis ; for his compositions seem to undulate, 
and he was characterized by the love of peregrina- 



122 POLLUX OR ERIDANUS. 

tion, and speaks of his expected pleasure in visit- 
ing the illustrious cities of Asia : Jam mens prce- 
trepidans avet vagari. His native city gave birth 
also to Paul Veronese, who, in his paintings, as- 
sembled the figures and dresses of strangers of 
all nations who came to Venice, and in this man- 
ner allied his productions to the agitation of exter- 
nal occurrences. It was said of Hannibal, the 
Carthaginian leader, that he generally did not fol- 
low out the advantages of the victory he gained ; 
and this trait corresponds with the want of in- 
clination to go on according to rule. But novelty 
must always result from there being two different 
positions of the same form to be contrasted, as ap- 
pears in the progression of this double form. 
Thus, the Lombardy poplar, growing beside the 
waters of Eridanus, when agitated by a breath of 
wind, shews the other sides of its leaves, having a 
different and lighter sort of green. 



123 



SECT. II. 

ON THE CONTINUED RELATION OF TWO SIMPLE 
CURVES. 

Castor or Better ophon. 

The application of the power of Neptune is capa- 
ble of producing also another double form, in 
which the two curves proceeding together are 
both simple, but in quality different from each 
other. Suppose that in the inner or ruling curve 
the same angle and the same proportion of lines 
is retained throughout, so as to continue always 
tracing the same form of a simple evolving hyper- 
bola, without successive branches ; and suppose 
that on the outside of this series of angles there 
be made to project from the points of the suc- 
cessive angles a series of lines diminishing accord- 
ing to the same proportion as that in which the 
straight lines guiding the inner curve increase ; 
then if the tops of these projecting lines be con- 
nected by other straight lines, an outer series of 
angles will be produced as before, and it will be 
capable of guiding an outer curve, which, instead 



124 CASTOR OR BELLEROPHON. 

of diverging from the other, will continually sink 
and approach towards it, because of the decreasing 
proportion of the lines extending between them. 
This outer curve will also be simple and without 
successive branches, because it is guided by a series 
of lines and angles generated always according to 
the same principle ; but the outer curve will not 
be the same form as the inner on a larger scale, 
but will be an hyperbolic form differing in quali- 
ty, because the successive quadrangles on which 
it is built are continually changing their forms 
and becoming more elongated ; because the up- 
right lines are shortening and decreasing in con- 
tinued proportion, while the bases and upper se- 
ries of lines are lengthening and increasing in con- 
tinued proportion. But all this proceeds accord- 
ing to an unchanging principle ; and the outer 
curve which is traced must be an hyperbola con- 
tinually sinking and approaching to the other. 
It is only as evolutes that two curves differing in 
their qualities as hyperbolic forms can continue 
to accompany each other as collateral forms ; be- 
cause their curvature is always becoming less, and 
bringing both of them nearer to the same standard, 
which is a straight line. If they were traced as 
involutes, in which the degree of curvature is al- 
ways increasing, their incongruity of direction 
would become more and more apparent ; and they 
would sooner or ]ater intersect, and cease to ex- 



CASTOR OR BELLEROPHON. 125 

tend as collateral forms accompanying each other. 
The reverse is the case in the conjoined progress 
of two evolutes approaching each other, and al- 
ways coming nearer to the same standard. But 
the breadth of the inner or lower curve would re- 
quire to be contracted in continued proportion, as 
in the power of Proserpina ; for otherwise the 
thickness of the curves would cause the two to 
meet, although the two series of angles guiding 
them remained always different and apart. In 
regard to length, their forms being evolutes, would 
continually increase ; and to prevent their exces- 
sive expansion, it becomes necessary that the forms 
always, after a certain number of angles, be trans- 
posed into a smaller scale, for instance a sixth. 
Thus, the inner curve, though simple, would be 
forced to break its continuity of extension, and 
assume, at each transposition, the form of a curve 
angle, having both sides convex ; and it is evi- 
dent also that at each diminution of scale the top 
of the last upright line (on the larger scale) would 
require to be connected with the top of the next 
(on the smaller scale) by a straight line making a 
great and sudden descent towards the inner curve^ 
the outer course being carried down along with 
it ; after which the form of the outer curve would 
again be produced as before, and would continue 
gradually sinking towards the inner curve. By 
means of the changes of course produced by these 



126 CASTOR OR BELLEROPHON. 

successive diminutions of scale, the general courses 
of the two curves together would be that of invo- 
lutes, passing into infinite refinement; although 
all the successive portions were parts of the same 
two evolutes. 

The peculiar characteristic of this power is, that 
the relation which has once begun to flow between 
the two different curves is always continued, and is 
drawn out like the waters of a smooth and placid 
river. It has relation to the pleasures of idleness, 
or of any simple and continuous employment ; and 
it was probably signified in the characters of the 
Sirens, who, on the coast of Sicily, endeavoured 
to detain Ulysses by their warbling. But it was 
probably also the true character of Achilles, the 
most powerful of the Grecian warriors ; because, 
by the diminution of scales, it contracts and 
compresses the future course of the same simple 
curves, and becomes as strength and swiftness. 
When Achilles withdrew in anger, because of the 
loss of Briseis, he was contented to remain idle 
in his tent. In his native region was the cele- 
brated vale of Tempe, watered by the Peneus, 
and affording such a prolongation of delight, that 
even the inhabitants of Olympus descended to 
waste their time in it. This mode of composition 
has most relation to the elementary power of Ceres ; 
because it is the continued approach of two dis- 
similar forms, between which the difference is al- 



CASTOR OR BELLEROPHON. 127 

ways becoming less, but cannot be exhausted. 
From the continued sinking of the outer curve, 
this power becomes the same as the love of hu- 
mility ; and, from never passing into a succession 
of different branches, it becomes the same as 
homeliness and contentment with things which are 
easily procured. This was probably the character 
of Antisthenes, the Cynic, and his follower Dio- 
genes, who studied to live on the cheapest possi- 
ble terms. In modern times, Swift was distin- 
guished for an affected humility in the choice of 
subjects for poetry ; Hogarth shewed the same 
taste in painting ; and Cobbett, the political writer, 
has endeavoured to illustrate the pleasures of 
thrift and of humble and contented labour, and 
to shew the value of the resources which are 
hidden in every plain material and vulgar pos- 
session. Similar characteristics are shewn in 
Cowper's mode of treating religious subjects in 
poetry. But all these persons were distinguished 
for those powers of hatred and recoil which belong 
to Ceres ; and they were characterized also by the 
inclination to study nature in its humblest forms, 
or to shew the resources contained in the simplest 
materials that can be assumed. This is also the 
character of Southey, among the modern English 
poets. To this mode of composition may be re- 
ferred the Elysian fields, where the shades of 
heroes existed in a state of pleasing idleness, 



128 CASTOR OR BELLEROPHON. 

amidst blooming meadows, receiving light from a 
sky of their own. Finding their happiness pro- 
longed from day to day, they wished for nothing 
more than to experience the progressive continua- 
tion of the same feelings. Among the nations of 
Europe, it would appear that some classes of the 
Spaniards approach the nearest to this character ; 
because they are distinguished for idleness and 
contentment with little, when more might easily 
be procured. Some travellers describe them as 
spending whole days wrapt in their cloaks, leaning 
in rows against a wall, or dozing under a tree. 
Sicily was considered as sacred to Proserpina, be- 
cause from thence she had been carried off by 
Pluto ; but Proserpina is the retrogression of one 
simple curve ; while the power of the Sirens is the 
evolution of two different simple curves, which to- 
gether are forced to assume the form of an invo- 
lute. Such is the freshness and continuity of 
vegetation, and, at the same time, its internal 
powers of repulsion or elasticity. 

But, in regard to the doubleness of the form, 
Thessaly was said to be also the country of the 
Centaurs. The education of Achilles was in- 
trusted to Chiron, in whose structure appeared 
the form both of horse and of man ; and this was 
probably meant to signify the conjoined progress 
of two curves altogether different in quality. To 
the same class may be referred Castor, the tamer 



CASTOR OR BELLEROPHON. 129 

of horses ; since the art of riding is skill in recon- 
ciling the powers and motions of two dissimilar 
forms. The same power belonged also to Belle- 
rophon, who mounted the winged Pegasus, and 
controlled him amidst the regions of the air. The 
name of Pegasus is derived from aw, a spring, 
because of the relation which extends like a river 
between the two different curves ; and something 
like this is the effort of the mind in continuous 
poetical invention, when it finds the materials al- 
together within itself. But to this mode of com- 
position must be referred the art of producing the 
bathos, or of finding something always more hum- 
ble, easy, and soothing to human nature, by shew- 
ing how much is contained in the most ordinary 
materials, and in those things which are not diffi- 
cult to be found. The Thessalians were cele- 
brated for their skill in enchantments, perhaps 
from the power of anticipating the future parts of 
the same continuity, and drawing them into a 
narrower and closer form, which is like anticipat- 
ing the future course of any simple affection of the 
mind, and increasing its strength by bringing al- 
ways more of it into a smaller compass. Castor, 
like Pollux, was represented with a star over his 
head ; and to him, more particularly, should be 
ascribed the power of tranquillizing the sea ; nor 
would Bellerophon have been thrown from his 
horse had he not attempted to soar to heaven, in- 

f 2 



130 CASTOR OR BELLEROPHON. 

stead of carrying Pegasus up into the clouds only 
to let him gradually and smoothly sink towards 
the terrestrial plains, where, the density of the air 
being increased, its elasticity would prevent him 
from ever reaching the ground. Such is the art 
of lowering a theme. Perhaps this was also the 
character of the Cumean or Campanian Sibyl, to 
whom Apollo granted an endless continuation of 
years ; but she, contracting and wearing away in 
her bodily form, seemed at last to retain only a 
voice. 



131 



CHAP. VII. 

ON THE POWERS OF COLLOCATION, OR THE 
DISTRIBUTION OF FINITE PARTS OF THE 
HYPERBOLA. 

Geryon or Silvanus. 

If the power of Vesta, which is equivalent to three 
angles or planes produced at the same time, be 
applied to the hyperbola, it must be supposed to 
separate it into three different parts or courses, 
each conducted by its own series of angles. But 
the power of Vesta is also that of terminating and 
breaking off, and it is also the power of a plane 
coming down to fasten upon the side of another, 
which gives the power of collocation or the idea of 
building ; and, therefore, in this mode of composi- 
tion, the series of angles which trace the curves 
may be supposed to break off and leave finite parts 
of the curves, but to retain the power of tracing 
out the remainders of the same curves elsewhere. 
In conformity to the power of Vesta, therefore, 
they may be supposed to build and fasten subse- 
quent portions of the same curves to the sides of 



• f 



132 GERYON OR SILVANUS. 

those portions which have already been placed; 
and to the sides of this second order of branches or 
terminated curves to add a third, and so on in con- 
tinued progression ; and the added parts being ap- 
plied to the points of angles, in the series which 
leads the former portion of the curve, the portions 
which are built upon that portion must be sup- 
posed to cross it. To enable this distribution of 
forms to proceed, it is evident that every succeed- 
ing order of parts must be transposed in a smaller 
scale, so as still to find room, and not be entangled 
with the others. On each finite portion of the 
curve two others must be built, that there may be 
a continued multiplication of parts. Thus, in a 
tree, each larger branch has smaller ones built 
upon it, and upon them are arranged smaller 
branches still. Without these transpositions, this 
mode of laying out the hyperbola would produce 
the grossest exuberance and confusion, which was 
probably signified in the character of the Hydra 
and the marsh of Lerna. But, by the continued 
diminution of scales, it assumes order and refine- 
ment ; and each portion of the curve, by also cross- 
ing within itself twice, may acquire variety of di- 
rection, and will not change the distribution of the 
general form. This mode of composition was pro- 
bably signified in Silvan us, the divinity of the 
woods, which have an obvious relation to the same 
form. The wife of Silvanus was named Fatua, 



GERY0N OR SILVANUS. 



138 



and was supposed to occasion the nightmare by 
perching on the breasts of persons asleep. But 
the character of Silvanus being founded on the 
power of spreading and distributing forms in every 
direction, and finding out new sites for them upon 
the sides of each other, may easily be distinguish- 
ed from the more vulgar attributes of Pan. The 
power of Silvanus is probably the true character 
of the people of Indostan, among whom is found 
the streaked tiger, a form of breaking off or tear- 
ing to pieces, as in the distribution of the parts of 
a single curve. Among the Indians are found the 
remains of the enormous systems of law, morality, 
and mythology. This mode of composition has 
most relation to the elementary power of Neptune, 
which bears two planes having relation to a com- 
mon basis ; and, in this mode of composition, each 
branch has two others built upon it. One of the 
ephithets of Neptune was Phutalmios, or produc- 
tive. To Silvanus may also be ascribed the power 
of melancholy and obscuration from intervening 
parts ; like the obscuration or gloom which exists 
in the midst of a forest. Nevertheless, this is the 
true character of the ship Argo, which signified 
the power of being loaded, or of carrying other 
forms upon it. This is the first type of naviga- 
tion ; and, among animals, the same character is 
expressed in the bear, which, in the northern seas, 
is sometimes found floating on large pieces of ice, 



134 GERYON OR SILVANUS. 

as if to signify its relation to loading. But a har- 
mony extends throughout the parts which are laid 
out according to this mode of composition ; because 
all the parts in each of the three curves has rela- 
tion to corresponding parts in the other two, at 
each transposition into a smaller scale, and through- 
out each new order of branches which is produced. 
The branches which correspond, however, are al- 
ways farther separated, and are forced to seek for 
each other through a crowd of intervening parts. 
This mode of composition was perhaps also signi- 
fied in three-bodied Geryon. Among the nations 
of modern Europe, this character belongs to the 
Germans in regard to obscurity and depth of feel- 
ing, and to the Dutch, in regard to methodical 
distribution, and the mere collocation of things not 
blended in the same continuity. Thus Silvanus 
differs from the power of Theseus. In another 
point of view, the power of Silvanus may be as- 
sumed as the orignial type of epic poetry, which 
is not a mere narrative of the order of events, but 
is rather an ample and profuse array of parts 
spreading in every direction, although all belong- 
ing to one subject or mode of deduction. To 
the same class with Silvanus therefore may be re- 
ferred the mode of construction which belongs to 
Homer, and which exemplifies the power of rami- 
fication and minuteness of detail, but still preserves 
the clearest order. This power is also shewn in 



GERYON OR SILVANUS. 135 

the Dutch school of painting, which expresses the 
nature of woods and marshes, and, in general, the 
various compilations and distributions of sub- 
stance. To this class may also be referred the 
science of geology, which had its origin among the 
Germans. Among the Homeric heroes this seems 
to have been the power belonging to the Sala- 
minian Ajax, who was considered as the next 
warrior after Achilles, but somewhat tardy and 
stupid. To this class may also be referred the 
philosophical system of Spinosa, who endeavoured 
to prove that the continuation of one substance, 
whether worldly or divine, constitutes the nature 
of all things, and extends throughout those forms 
which have the appearance of being separate and 
particular. This mode of composition was, per- 
haps, also signified in the golden fleece, which was 
the object of the expedition undertaken by Jason. 
The artificial use of substance was expressed in 
the poisons and enchantments of Medea, and also 
in her cutting GEson to pieces to restore him to 
youth, which applies to breaking off the hyperbola 
and laying it out in a different order. The golden 
ram of Phryxus was capable of transporting its 
owner through the air, because it carries parts of 
the same curve into a new order. To this class 
may perhaps be referred the artist Daedalus, whose 
wings, being constructed only with wax, could not 
bear the heat of the sun. The power of Silvanus 



136 GERYON OR SILVANUS. 

is departure from continued derivation in joining 
parts of the same form together ; and to Silvanus 
may be ascribed sententiousness, which gives forth 
wisdom in minute portions. This character be- 
longed, among the English poets, to Pope; but 
the same power appeared in Virgil as art in shift- 
ing the arrangement of his words, and producing 
beauties of collocation. But to Silvanus may also 
be ascribed the study of conic sections, as the 
means of comparing different parts of the same 
curve. Mathematical reasoning depends much on 
the power of arbitary collocation and comparison 
of figures. Sir Isaac Newton, whose genius was 
gross and tardy, like that of Ajax, was a native of 
Lincolnshire, a region abounding with fens. From 
each two later branches being fastened upon the 
side of a former portion of the same simple curve, 
this mode of composition is like a reference to per- 
sonal experience or past states of the mind. This 
constitutes the sedateness of age. The essential 
characteristic of a man of experience and sententi- 
ous wisdom is the power of referring to situations 
in which he has formerly been. Thus the genius 
of Newton fitted him for comparing the different 
portions of the trajectories or courses through which 
moving bodies pass. But this mode of composi- 
tion, from each new form being speedily broken 
off and left terminated, becomes like a feeling of 
the vanity of all human undertakings, and gives 



GERYON OR SILVANUS. 137 

birth to a peculiar strain of morality. Thus Solo- 
mon, having tried every different pursuit and enjoy- 
ment, and having accumulated vast experience as 
an individual, declared all things to be vanity, a 
declaration which applied only to his own kind of 
progression as to feeling. This mode of writing 
is studied by Lord Byron among the modern Eng- 
lish poets. To Silvanus, however, belongs a great 
depth and tenderness of sentiment, founded always 
on the reference of present affections to those 
which have preceded them. Thus the wallflower, 
egregious in its tints, nods over the remains of 
some dilapidated tower. At each new distribu- 
tion of forms, by the power of Silvanus, the forms 
brought together are more remote as to their real 
places in the curve from whence they are taken. 
Thus the stork and crane, and other aquatic birds, 
unite in themselves the substances collected from 
remote marshes. To the same class may be re- 
ferred the literary commentators who build their 
observations upon those of previous commentators 
till the structure becomes overloaded. But it is 
evident that the successive orders of forms, pro- 
ceeding according to the power of Silvanus, are 
capable of infinite progression. This power has a 
certain relation to the finite ranges successively ex- 
plored by the power of Orion or of Bacchus, who 
was celebrated for having conquered India ; and, 
perhaps, his attendant Silenus was in reality a form 



138 GEEYON OE SILVANUS. 

of Silvanus. But as to the accumulation of parti- 
cular forms, this power was evidently signified in 
the character of Plutus, the god of wealth, who de- 
rived his origin from Jason and Ceres ; for every 
little portion which is broken off is gained from an 
infinite curve, which can never be exhausted. 

After this power of Silvanus, which is produced 
from the application of the power of Vesta, this 
series is done ; because the next power, which is 
Mars, is the same as that by which the hyperbola 
was originally accelerated. Therefore, after this, 
there is only room for a composition of composi- 
tions,. 



AN 



INDEX 



PROPER NAMES 



MENTIONED IN THE FOREGOING TREATISE, 



A 


PAGE 




PAGE 


Acheron 


■ 83 


Ashur, tribe of 


■ 34 


Achilles 


126 


Athenians 


108 


Adonis 


81 


Atlas 


120 


iEsculapius 


67 






Ajax 


134 


B 




Alcaeus 


54 


Bacchus 


49 


Anacreon 


63 


Bacon 


64 


Angelo, Michael 


- 78 


Bellerophon 


123 


Antisthenes 


127 


Benjamin, tribe of 


45 


Apollo 


15 


Bunyan 


120 


Arcadians 


84 


Buonaparte 


71 


Argives 


96 


Byron 


137 


ArgOj the ship 


131 






Ariadne 


65,77 


C 




Aristophanes 


104 


Cadmus 


104 


Aristotle 


66 


Caelus 


6 



140 


INDEX. 






PAGE 




PAGE 


Carthaginians 


121 


Elysian fields 


127 


Castor 


123 


English 


64 


Catullus 


121 


Ephraim 


115 


Centaurs 


128 


Epicurus 


115 


Ceres 


- 38 


Erichthonius 


100 


Chaldeans 


- 93 


Eridanus 


119 


Chimaera 


116 


Eros 


80 


Chinese 


105 


Etrurians 


78 


Chiron 


128 






Circe 


87 


F 




Cobbett 


129 


Fates 


94 


Cocytus 


89 


Fatua 


132 


Coleridge 


92 


Fauns 


58 


Cowper 


127 


Flora 


78 


Crete 


65 


French 


109 






Furies 


89 


D 








Daedalus . - 


135 


G 




Dan, tribe of 


25 


Gad, tribe of 


29 


Dante 


78 


Galileo 


78 


Democritus 


109 


Ganymede 


105 


Demogorgon 


88 


Germans 


134 


Diana 


18 


Geryon 


131 


Diogenes 


127 


Goethe 


80 


Diomed 


57 


Graces, the 


53 


Dutch 


134 


H 




E 




Handel 


121 


Egyptians 


102 


Haydn 


113 


Elis 


■ 76 


Hebe 


56 





INDEX. 


141 




PAGE 




PAGE 


Hercules 


60 


Locke 


105 


Hesiod 


80 


Lucifer 


19 


Hogarth 


127 






Homer 


134 


M 




Horace - , 


93 


Mars 


29 


Hume, David 


104 


Matthew, St 


106 


Hydra 


132 


Medea 


135 


Hymen 


121 


Medusa 


89 






Meleager 


54 


I 




Mercury 


45 


Jacob - » 


. 6 


Milton 


86 


■Jason 


135 


Minerva 


41 


Jeffrey 


109 


Minos 


65 


Jerusalem 


93 


Morpheus 


63 


Indostan 


131 


Moses 


103 


Johnson, Samuel - 


109 


Muses, the three earlier 11 


Joseph 


42 


Muses, the nine 


60 


Irish 


121 


Mytelene 


54 


Issachar, tribe of - 


25 






Italy 


55 


N 




Judah, tribe of 


18 


Naphthali 


38 


Judas Iscariot 


66 


Nemesis 


93 


Juno 


5 


Neptune 


25 


Jupiter 


9 


Nestor 


80 






Newton 


136 


L 




Niobe 


105 


Lacedaemonians 


90 






Latona 


15 


O 




Lethe 


96 


Olympic games 


76 


Levi, tribe of 


16 


Oreades 


111 



142 



INDEX. 





PAGE 


R 


PAGE 


Orion 


49 


Raphael 


84 


Ovid 


103 


Reuben, tribe of 


. 6 






Rhea 


26 


P 




Rhodians 


104 


Palamedes 


96 


Romans 


55 


Pan 


67 


Rousseau 


104 


Paul, St - 


66 


Rubens 


84 


Pegasus 


129 






Pelops 


72 


S 




Perseus 


88 


Sappho 


54 


Persians 


86 


Saturn 


46 


Phlegethon 


75 


Scotch 


69 


Phoenicians 


97 


Scott, Sir Walter - 


121 


Pindar 


96 


Shakspeare 


112 


Pirithous 


87 


Sibyl 


130 


Plato 


102 


Sicily 


128 


Pluto 


81 


Silenus 


137 


Plutus 


138 


Silvanus 


131 


Po, the river 


119 


Simeon, tribe of 


10 


Pollux 


117 


Sirens 


126 


Pope, the 


57 


Socrates 


112 


Pope, the poet 


136 


Solomon 


137 


Pomona 


57 


Somnus 


91 


Priapus 


56 


Southey 


127 


Prometheus 


69 


Spaniards 


128 


Proserpina 


72 


Spartans 


90 


Proteus 


26 


Spenser 


63 


Pythagoras 


86 


Spinosa 


135 


Pythian Apollo 


69 


Styx 


86 





1IN1 

PAGE 


uiijL* 


PAGE 


Swift 


127 


Uranus 


6 


Swiss 


104 


V 




T 




Venetians 


97 


Tempe 


126 


Venus 


34 


Thales 


97 


Veronese 


122 


Thamyris 


79 


Vertumnus 


94 


Thebans 


103 


Vesta 


24 


Themis 


12 


Virgil 


136 


Theophrastus 


54 


Voltaire 


113 


Theseus 


107 


Vulcan 


20 


Thessaly 


128 






Typhon 


102 


W 








Wordsworth 


78 


U 








Ulysses 


120 


Z 




Urania 


8 


Zabulon, tribe of 


- 21 



THE END, 



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